Let $(u_1, u_2, u_3, u_4)$ and $(v_1, v_2, v_3, v_4)$ be vectors in $\mathbb R_+^4$. Is the following inequality true?

\begin{align*} \left(\sum_{{[4] \choose 3}} \sqrt{u_i u_j u_k}\right)^{2/3} + \left(\sum_{{[4] \choose 3}} \sqrt{v_i v_j v_k}\right)^{2/3} \leq \left(\sum_{{[4] \choose 3}} \sqrt{(v_i+u_i) (v_j+u_j) (v_k+u_k)}\right)^{2/3} \end{align*} Here $\sum_{{[4] \choose 3}}$ refers to $\sum_{1\leq i < j < k \leq 4}$.

I ran a million Matlab simulations for random vectors and it did not yield any counterexample.

Note: Previously asked on MSE (https://math.stackexchange.com/questions/2647608/a-minkowski-like-inequality-for-symmetric-sums)