Zhang's unitary-orbit $n$-tuple Schatten inequality conjecture

Let $A_1,\ldots,A_n\in\mathbb{M}_m$, and let $|A|$ denote the positive operator $(A^*A)^{1/2}$. Zhang's conjecture. For $p>2$, there exist $\frac{n(n-1)}{2}+1$ unitary matrices $U$ and $U_{i,j}$, with $1\le i<j\le n$, such that

$$U\left|\sum_{i=1}^n A_i\right|^pU^*+\sum_{1\le i<j\le n}U_{i,j}\left|A_i-A_j\right|^pU_{i,j}^*\le n^{p-1}\sum_{i=1}^n|A_i|^p.$$

For $0<p\le2$, the inequality is reversed. This is described as a tuple generalization and strengthening of an earlier operator inequality; no resolution is supplied in the source.