The closed-form conjecture for the optimal Schatten norm constant for m matrices

Let $M_n(\mathbb C)=\mathbb C^{n\times n}$, and for $A\in M_n(\mathbb C)$ write $|A|=(A^*A)^{1/2}$. For $p>1$ and $m\geq2$, let $c_p(m)$ be the smallest number such that

$$\left\|\sum_{k=1}^m A_k\right\|_p\leq c_p(m)\left\|\sum_{k=1}^m|A_k|\right\|_p$$

for all $A_1,\dots,A_m\in M_n(\mathbb C)$. The closed-form conjecture. If $x_{p,m}>0$ is the positive solution of

$$x^p-2x-(m-1)=0,$$

then

$$c_p(m)=\frac{\sqrt{x_{p,m}(x_{p,m}+m-1)}}{\left(x_{p,m}^{\,p}+m-1\right)^{1/p}}.$$

This conjecture extends Lee's problem from two matrices to an arbitrary number of matrices and is motivated by numerical experiments. The paper notes that the formula recovers the known cases $p=1,2,\infty$, while the optimal value for general $p>1$ is the unresolved issue addressed by the conjecture.