Hanner inequality for Schatten norms of operator tuples

Let $A_0,A_1,\ldots,A_{n-1}$ be operators on a separable Hilbert space, and let $p,q$ satisfy $1/p+1/q=1$. Operator Hanner-type conjecture. For $2\leq p<\infty$,

$$n\left(\sum_{j=0}^{n-1}\lVert A_j\rVert_p^p\right)^{q/p}\leq\left\lVert\sum_{j=0}^{n-1}A_j\right\rVert_p^q+\sum_{0\leq j<k\leq n-1}\lVert A_j-A_k\rVert_p^q,$$

whereas for $0<p\leq2$,

$$\left\lVert\sum_{j=0}^{n-1}A_j\right\rVert_p^q+\sum_{0\leq j<k\leq n-1}\lVert A_j-A_k\rVert_p^q\leq n\left(\sum_{j=0}^{n-1}\lVert A_j\rVert_p^p\right)^{q/p}.$$

The conjecture is presented as a natural extension of known operator inequalities and would generalize Hanner-type inequalities; the source gives no resolution.