Bourin–Lee's multivariable unitary-orbit inequality conjecture

Let $A_1,\ldots,A_n$ be $m\times m$ matrices and let $p>2$. There are $\frac{n(n-1)}{2}+1$ unitary matrices $U$ and $U_{i,j}$ for $1\le i<j\le n$.

Bourin–Lee's multivariable conjecture. The unitary matrices can be chosen so that

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

For $0<p\le 2$, the inequality should be reversed. This is proposed as an $n$-variable extension of a known two-variable matrix inequality of Bourin and Lee; the source does not provide a resolution, so the conjecture remains open.