Noncommutative Clarkson inequality for Schatten norms

Let $\mathscr{H}$ be a Hilbert space and let $\mathbb{B}_p(\mathscr{H})$ denote the Schatten $p$-class on $\mathscr{H}$. For $A,B\in\mathbb{B}_p(\mathscr{H})$, the noncommutative Clarkson conjecture. For $p\ge2$,

$$2\bigl(\|A\|_p^2+\|B\|_p^2\bigr)^{p/2}\le\|A+B\|_p^p+\|A-B\|_p^p,$$

equivalently,

$$\|A\|_p^2+\|B\|_p^2\le\left(\frac{\|A+B\|_p^p+\|A-B\|_p^p}{2}\right)^{2/p}.$$

For $0<p\le2$, the inequality is reversed. It is motivated by the scalar convexity inequality and its known operator analogue; the supplied text does not state whether this conjecture has been resolved.