Conjecture of van der Holst and Pendavingh related to bound for Colin de Verdière invariant

Kaluza and Tancer have actually proved $\mu(G)\leq\sigma(G)$ in 2019: See their proof in the preprint "Even maps, the Colin de Verdière number, and representations of graphs" on arxiv. Here is the link https://arxiv.org/pdf/1907.05055.pdf

You are right, the invariant $\sigma(G)$ of Holst and Pendavingh does not seem to have an established name yet.

I would like to add one important aspect: it was known that $\mu(G)$ and $\sigma(G)$ can deviate by a large amount for larger values $k$. Now we have the proof of the improved (sharper) bound $\mu(G)\leq\sigma(G)$, but even though this is an improvement, Kaluza and Tancer also showed that a large gap exists already for small values of $k$: They showed there is a graph $G$ such that $\mu(G)\leq7$ and $\sigma(G)\geq8$ ("Even maps, the Colin de Verdière number, and representations of graphs" on arxiv. Here is the link https://arxiv.org/pdf/1907.05055.pdf).
Now, a suspension of $G$ (adding a new vertex to $G$ and connecting it to all vertices of $G$) increases both $\mu(G)$ and $\sigma(G)$ by exactly one (unless $G$ is the complement of $K_2$). Therefore $\mu(G)\leq7$ and $\sigma(G)\geq8$ implies that for every $k \in\mathbb N$, $k \geq 7$, there must exist a graph $G_k$ with $\mu(G)\leq k$ and $\sigma(G)\geq k+1$, i.e. strict inequality for all large values of $k$. Finally, the authors also show that the gap between $\mu(G)$ and $\sigma(G)$ is asymptotically large.