The norm inequality for dimensions of projective indecomposable modules

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

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

Let $\overline{\varphi}=(\varphi_1(1),\ldots,\varphi_k(1))$ be the vector of irreducible $p$-Brauer character degrees, and let $c=(c_{\varphi_1},\ldots,c_{\varphi_k})$ be the vector of dimensions of the corresponding projective indecomposable modules. Equip $\mathbb{R}^k$ with its Euclidean scalar product and norm.

The norm inequality. The inequality

$$\lVert c\rVert\leq\lVert\overline{\varphi}\rVert$$

holds for every finite group.

The inequality compares the Euclidean norms of the dimensions of projective indecomposable modules and irreducible $p$-Brauer characters. The source presents it as a conjecture based on examples; no resolution is supplied here.