The Schatten norm conjecture for r-partite graphs

Let $p>2$, $n\geq r\geq2$, and let $G$ be an $r$-partite graph of order $n$. Let $T_r(n)$ be the Turán graph on $n$ vertices.

Schatten norm conjecture for r-partite graphs. If $G$ is an $r$-partite graph of order $n$, then

$$\|G\|_p<\|T_r(n)\|_p,$$

unless $G=T_r(n)$.

The analogous assertion is stated as an elementary extremal result for $p=2$. For $p>2$, the paper presents this strict Turán extremality statement as plausible, while the preceding constructions provide lower-bound evidence relevant to the problem.