Audenaert–Kittaneh's Clarkson inequalities conjecture for Schatten operators

Let $4A_1,\ldots,A_nin \mathbb{B}_p(\mathscr{H})$. Let $q>0$ satisfy

$$\frac{1}{p}+\frac{1}{q}=1.$$

Audenaert–Kittaneh's conjecture. For $1<p\le 2$,

$$\left\| \sum_{i=1}^nA_i\right\| _p^q+\sum_{1\le i<j\le n}\|A_i-A_j\|_p^q\le n\left( \sum_{i=1}^n\|A_i\|^p_p\right)^\frac{q}{p},$$

and for $p\ge 2$,

$$n\left( \sum_{i=1}^n\|A_i\|^p_p\right)^\frac{q}{p}\le \left\| \sum_{i=1}^nA_i\right\| _p^q+\sum_{1\le i<j\le n}\|A_i-A_j\|_p^q.$$

The conjecture is a several-operator extension of the Clarkson–McCarthy inequalities. The paper proves the first inequality for $1<p\le 2$ and obtains the reverse inequality for $2\le p<\infty$ from a duality theorem, so the conjecture is solved.