The strict Schatten norm conjecture for graphs of fixed order

Let $p>2$ and let $G$ be a graph of order $n$. Let $K_n$ be the complete graph on $n$ vertices.

Strict Schatten norm conjecture. If $p>2$ and $G$ is a graph of order $n$, then

$$\|G\|_p^p<\|K_n\|_p^p,$$

unless $G=K_n$.

The preceding argument gives an asymptotically tight order-based upper bound whose equality is never attained; this conjecture asserts that the complete graph is the unique maximizer for every $p>2$.