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on Willems' lower bound for Brauer character degrees

The statement

Let $G$ be a finite group and $p$ a prime. Let $\operatorname{IBr}_p(G)$ be the set of irreducible $p$ -Brauer characters of $G$ , and let $|G|_{p'}$ be the largest divisor of $|G|$ coprime to $p$ . Is $\sum_{\varphi \in \operatorname{IBr}_p(G)} \varphi(1)^2 \ge |G|_{p'}$ always true?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.