Strengthened Fourier-type $n$-operator Clarkson inequality conjecture

Let $\mathscr{H}$ be a Hilbert space, let $\mathbb{B}_p(\mathscr{H})$ denote the Schatten $p$-class, and define $\omega_j=e^{ji\pi/n}$. Let $A_1,\ldots,A_n\in\mathbb{B}_p(\mathscr{H})$. The strengthened Fourier-type conjecture. For $p\ge2$,

$$n^{2-p/2}\left(\sum_{j=1}^n\|A_j\|_p^2\right)^{p/2}\le\sum_{k=1}^n\left\|\sum_{j=1}^n\omega_{j-1}^{k-1}A_j\right\|_p^p.$$

For $0<p\le2$, the inequality is reversed. The source introduces this as a replacement conjecture after reporting that the preceding Fourier-type conjecture is false; no resolution of the strengthened assertion is supplied.