The sharp Schatten norm conjecture for graphs with fixed size

Let $p>2$ and let $G$ be a graph with $m$ edges. Write $\|G\|_p$ for the Schatten $p$-norm of its adjacency matrix.

Sharp Schatten norm conjecture. If $p>2$ and $G$ is a graph of size $m$, then

$$\|G\|_p^p\leq\left(-\frac{1}{2}+\sqrt{2m+\frac{1}{4}}\right)^p-\frac{1}{2}+\sqrt{2m+\frac{1}{4}}.$$

Equality holds if and only if $G$ is a complete graph with possibly some isolated vertices.

The proposition preceding the conjecture gives a tight asymptotic upper bound, while equality in that bound never holds; this conjecture proposes the exact finite-size bound and characterizes its equality cases.