A (reverse)-Minkowski type inequality for symmetric sums
Rewrite the inequality in question as \begin{equation*} f(u+v)\le f(u)+f(v) \end{equation*} for $u,v$ in $\mathbb R_+^4$, where \begin{equation*} f(u):=-\left(\left(\frac{1}{\sqrt{u_1}}+\frac{1}{\sqrt{u_2}}+\frac{1}{\sqrt{u_3}} +\frac{1}{\sqrt{u_4}}\right) \sqrt{u_1 u_2 u_3 u_4}\right)^{2/3}. \end{equation*} Note that the function $f$ is positive homogeneous: $f(tu)=tf(u)$ for $t\ge0$. So, $f(u+v)=2f(\frac{u+v}2)$.
It remains to notice that $f$ is convex. Indeed, the determinant of the Hessian matrix \begin{equation} M:=\Big(\frac{\partial^2 f}{\partial u_i\partial u_j}\Big)_{i,j=1}^4 \end{equation} is $0$, and the principal minors of $M$ are manifestly positive, after some algebraic simplifications. (That the determinant of $M$ is $0$ can be shown either by direct calculations or by recalling that $f$ is positive homogeneous and hence $\frac{d^2}{dt^2}\,f(tu)=0$ for $t>0$.)
Dealing with the determinants of the matrices $\Big(\frac{\partial^2 f}{\partial u_i\partial u_j}\Big)_{i,j=1}^k$ for $k=1,2,3$, in view of the positive homogeneity, we may assume without loss of generality that $u_4=1$.
Details of the calculations can be seen in the the Mathematica notebook and/or its pdf image .
The said claim follows from the following general result on elementary symmetric polynomials, denoted $e_k$ below. $\newcommand{\vx}{\mathbf{x}}\newcommand{\vy}{\mathbf{y}}$
Theorem A (S. 2018). $\,$ Let $p\in (0,1)$ and $x \in \mathbb{R}_+^n$. The map \begin{equation*} \phi_{k,n}(x) := x \mapsto \left[\frac{e_k(x_1^p,\ldots,x_n^p)}{e_{k-1}(x_1^p,\ldots,x_n^p)}\right]^{1/p}, \end{equation*} is concave. Recently, I typed up a proof to this inequality (and a few others). Please see this preprint for a proof.
As a corollary, we obtain a concavity result that implies the OP's conjectured inequality as a special case (using positive homogeneity).
Corollary. Let $p\in (0,1)$ and $\vx \in \mathbb{R}_+^n$. Then, $[e_k(\vx^p)]^{1/pk}$ is concave (we write $\vx^p \equiv (x_1^p,\ldots,x_n^p)$. \begin{align*} [e_k((\vx + \vy)^p)]^{1/pk} &= \left[\frac{e_k((\vx+\vy)^p)}{e_{k-1}((\vx+\vy)^p)}\cdot \frac{e_{k-1}((\vx+\vy)^p)}{e_{k-2}((\vx+\vy)^p)}\cdots \frac{e_1((\vx+\vy)^p)}{e_0((\vx+\vy)^p)} \right]^{1/pk}\\ &= \left[\phi_{k,n}(\vx+\vy)\phi_{k-1,n}(\vx+\vy)\cdots\phi_{1,n}(\vx+\vy)\right]^{1/k}\\ &\ge \left[\left(\phi_{k,n}(\vx)+\phi_{k,n}(\vy)\right)\cdots\left(\phi_{1,n}(\vx)+\phi_{1,n}(\vy)\right)\right]^{1/k}\\ &\ge \prod_{j=1}^k[\phi_{j,n}(\vx)]^{1/k} + \prod_{j=1}^k[\phi_{j,n}(\vy)]^{1/k}\\ &= [e_k(\vx^p)]^{1/pk} + [e_k(\vy^p)]^{1/pk}, \end{align*} where the first inequality follows from Theorem A and the second is just Minkowski.