Inequality for hook numbers in Young diagrams
Not sure, please check carefully. (Well, now more sure and the argument is more direct.)
I claim that the array $(h)$ majorates the array $(q)$, that is, $\sum \varphi (h_{ij})\geqslant \sum \varphi(q_{ij})$ for any convex function $\varphi$, in particular for $-\log$, that is your inequality.
Denote the hook lengths of the first (largest) column by $0<c_1<c_2<\dots<c_m$. Then the hooks in $i$-th row, which contains $c_i-i+1$ squares, are all numbers from 1 to $c_i$ except $c_i-c_1$, $c_i-c_2$, $\dots$, $c_i-c_{i-1}$ (this elementary claim is well-known, it shows the equivalence of Frobenius and the hook length formulae.) That is, $$ A:=\sum \varphi(h_{ij})=\sum_i \bigl(\varphi(1)+\ldots+\varphi(c_i)\bigr)-\sum_{i<j} \varphi(c_j-c_i). $$ Clearly $$ B:=\sum \varphi(q_{ij})=\sum_i \bigl(\varphi(m-i+1)+\ldots+\varphi(c_i+m-2i+1)\bigr). $$
For $i<j$, we have: $$ \varphi(j-i)+\varphi(c_j-i+1)\geqslant \varphi(c_j-c_i)+\varphi(c_i+j-2i+1), $$ since for $c_j=c_i+j-i$ the equality takes place, and the difference of two sides increases as a function of $c_j\in [c_i+j-i,+\infty)$. Summing up these inequalities over all pairs $i<j$ we get the desired inequality $A-B\geqslant 0$.
If $\varphi$ is strictly convex, then all inequalities are sharp only for a rectangle.
I was recently thinking about this question again and I came up with another way to explain the result $$\mathcal H_{\lambda}=\left\{ h(i,j)\right\}_{(i,j)\in \lambda}\succcurlyeq \left\{ h^{*}(i,j)\right\}_{(i,j)\in \lambda}=\mathcal H^{*}_{\lambda} \tag{1}$$ in a way that (i) combinatorially describes the sequence of Robin Hood moves to get from $\mathcal H_{\lambda}$ to $\mathcal H^{*}_{\lambda}$ and (ii) gives bijective proofs of the various inequalities that follow. In particular there is a bijective explanation for the original inequality: $$\prod_{(i,j)\in \lambda} h(i,j)\le \prod_{(i,j)\in \lambda} h^{*}(i,j) \tag{2}$$
Majorization and RH moves: Suppose that $\mu, \nu$ are two integer partitions of $n$. We say that $\mu\succcurlyeq \nu$ if $$\sum_{i=1}^{r}\mu_i\geq \sum_{i=1}^r \nu_i $$ for all $r\geq 1$ (if the partitions do not have the same number of parts, we can add extra zero entries at the end).
A Robin Hood (RH) move on $\mu$ is to pick to parts $\mu_i> \mu_j$ and replace them with $\mu_i -1, \mu_j+1$ and reorder the terms. This can be pictured on the associated Young diagram as the move which removes a single square at the end of a row and attaches it at the end of some row of shorter length while preserving the property of being a partition. These and their inverse also more frequently known as lowering/raising operators, I just like the metaphor of giving to the less fortunate parts of the partition.
A fundamental fact is that RH moves describe the covering relation in the majorization poset of partitions. In other words, $\mu\succcurlyeq \nu$ is equivalent to being able to obtain $\nu$ from $\mu$ by successively applying RH moves. When working with majorization in combinatorial settings this description is often more desirable than the analytic description (such as when working with symmetric functions, like in Macdonald's book, for example).
Partitions as directed graphs: Given a partition $\lambda=(\lambda_1,\dots,\lambda_k)$, its Young diagram can also be viewed as a directed graph $G(\lambda)$ by taking the vertices to be the lattice points $(i,j)$ with $1\le i\le \lambda_j$ and $1\le j\le k$, and edges to be $(i_1,j)\to (i_2,j)$ when $i_2\geq i_1$ and $(i,j_1)\to (i,j_2)$ when $j_2\geq j_1$. Notice that equality is allowed, so there is a self loop at every vertex. The graph $G^{*}(\lambda)$ is defined the same as $G(\lambda)$ but with all the edges reversed.
The set of outdegrees of $G(\lambda)$ is precisely $\mathcal H(\lambda)$ and similarly the outdegrees of $G^{*}(\lambda)$ for the set $\mathcal H^{*}(\lambda)$. We will think of the multisets $\mathcal H(\lambda),\mathcal H^{*}(\lambda)$ as partitions of $|\lambda|+N$ where $$N= \sum_{i=1}^{k}\binom{\lambda_i}{2}+\sum_{i=1}^{r}\binom{\lambda'_i}{2}.$$ Here $\lambda'$ is the conjugate partition of $\lambda$.
We can put a total order on the edges of $G(\lambda)$ that aren't loops by making all horizontal edges come before vertical edges and the edge $(i_1,j_1)\to (i_2,j_1)$ comes before $(i_3,j_2)\to (i_4,j_2)$
with a similar order on the vertical edges. Therefore we can refer to the nonloop-edges in order as $\{e_1,e_2,\dots,e_N\}$.
We can define the sequence of directed graphs $\{G(t)\}_{t=1}^{N}$ by $G(1)=G(\lambda)$ and obtaining $G(k+1)$ from $G(k)$ by reversing the direction of edge $e_k$. The final graph is $G(N)=G^{*}(\lambda)$. Let $H(t)$ denote the partition of $|\lambda|+N$ given by the outdegrees of $G(t)$.
Theorem: The partition $H(t+1)$ is obtained from $H(t)$ by an RH move.
Proof: $H(t+1)$ differs from $H(t)$ simply by the reversal of edge $e_t:u\to v$. We see that the parts $(\deg(u),\deg(v))$ get sent to $(\deg(u)-1,\deg(v)+1)$. So we simply have to check that $\deg(u)>\deg(v)$ in $G(t)$, but this follows by checking that for every edge $v\to w$ we have and edge $u\to w$ together with the fact that there is no edge $v\to u$.
Corollary (1): We have $\mathcal H_{\lambda}\succcurlyeq \mathcal H^{*}_{\lambda}$. Moreover $H(N_0)$ for $N_0=\sum_{1}^k\binom{\lambda_i}{2}$ recovers the multiset of semi-hooks from the Pak-Petrov-Sokolov paper, and we have $$H(0)\succcurlyeq H(N_0)\succcurlyeq H(N).$$
Bijective proof of (2): The quantity $\prod_{(i,j)\in \lambda} h(i,j)$ counts the number of subgraphs of $G(\lambda)$ of outdegree 1 at every vertex.The inequality follows if we can describe an injection from the set of such subgraphs to the subgraphs of outdegree 1 of $G^{*}(\lambda)$.
Start with a subgraph of outdegree 1 of $G(1)$ and start reversing the edges one by one according to the order defined previously producing an outdegree 1 subgraph of $G(t)$ for every $t$. When we reverse the edge $u\to v$ we also replace the outgoing edge $v\to w$ by $u\to z$ where $z$ is such that $v+z=u+w$ as vectors (in other words, $z,w,v,u$ are the vertices of a (possibly degenerate) rectangle). This ensures that the subgraph remains of outdegree 1.
Notice that this is an injection because it can be performed backwards in a unique way whenever it is possible. When the partition is not a rectangle and we try to run the process backwards, the solution to $v+z=u+w$ may take us to a square outside of $\lambda$ so we conclude that our map will be a strict injection in that case.
Note: The injection defined above may seem a bit arbitrary but it is simply combining the RH moves with the bijective proof of $$(a-1)(b+1)\geq ab$$ for some integers $a>b$.