The PSD block-sum Schatten maximization conjecture

Let $Q^{(i)}=[Q^{(i)}_{(jk)}]_{j,k=1}^2$ be arbitrary positive semidefinite $2\times2$ block matrices, and let $q^{(i)}_k$ be arbitrary non-negative numbers. For a matrix $A$, write $\lVert A\rVert_q$ for its Schatten $q$-norm. The PSD block-sum Schatten maximization conjecture. Under the constraints

$$\lVert Q^{(i)}_{(kk)}\rVert_q=q^{(i)}_k,$$

the maximum attainable value of $\left\lVert\sum_iQ^{(i)}\right\rVert_q$ is

$$\left\lVert \begin{pmatrix} \sum_i q^{(i)}_1 & \sum_i\sqrt{q^{(i)}_1q^{(i)}_2}\\ \sum_i\sqrt{q^{(i)}_1q^{(i)}_2} & \sum_i q^{(i)}_2 \end{pmatrix} \right\rVert_q.$$

This is presented as an equivalent formulation of the proposed norm-compression inequality in the case $p=2q$. The supplied text does not establish its resolution.