Monotonicity of the function $F_p$ under orthogonalization

Let $U_N^N$ be the space of orthogonal matrix models, let $K_N$ be the subspace of magic matrices, and let $\Psi$ be the orthogonalization map. For each fixed integer $p\geq2$, define

$$F_p(x)=\frac{1}{N^p}\left\|T_p^x\xi_{\sqcap\hskip-0.5mm\sqcap\ldots\sqcap}\right\|^2.$$

$F_p$-monotonicity conjecture. For every $x\in U_N^N$ and every $p\geq2$,

$$F_p(x)\geq F_p(\Psi^2(x)),$$

with equality if and only if $x\in K_N$, in which case $F_p(x)=1$. The source reports computer evidence for this statement and notes that compactness would make it imply convergence of the Sinkhorn-type algorithm.