Sharp three-operator Clarkson–McCarthy inequality

Let $S_p$ denote the Schatten $p$-class, and let $A,B,C\in S_p$. For $p>0$, write $\\|\cdot\\|_p$ for the Schatten $p$-quasi-norm. Sharp three-operator Clarkson–McCarthy conjecture. For $p\ge2$,

$$\\|A+B+C\\|_p^p+\\|A\\|_p^p+\\|B\\|_p^p+\\|C\\|_p^p \le \frac{3^{p-1}+1}{2^p}\left(\\|A+B\\|_p^p+\\|B+C\\|_p^p+\\|C+A\\|_p^p\right).$$

For $0<p\le2$, the inequality is reversed. The constant $\frac{3^{p-1}+1}{2^p}$ is optimal, with equality for all $p>0$ when $A=B=C\ne0$. This is proposed as a sharp complement to the preceding Clarkson–McCarthy-type theorem; a weaker bound is known, but the sharp inequality remains unproved in the source.