The Brauer character degree-square inequality and normal Sylow subgroup criterion

Let $G$ be a finite group and let $p$ be a prime. Write

$$\operatorname{IBr}_p(G)=\{\varphi_1,\ldots,\varphi_k\}$$

and let $\overline{\varphi}=(\varphi_1(1),\ldots,\varphi_k(1))$ be the vector of irreducible $p$-Brauer character degrees. Let $|G|_{p'}$ denote the part of $|G|$ coprime to $p$, and let a Sylow $p$-subgroup mean a subgroup of $G$ of maximal $p$-power order.

The Brauer degree-square conjecture. One always has

$$|G|_{p'}\leq\lVert\overline{\varphi}\rVert^2,$$

with equality if and only if the Sylow $p$-subgroup is normal.

The conjecture was proposed in earlier work cited by the source and is motivated by examples. The supplied text does not state a proof or a resolution.