Conjecture B on the number of Brauer constituents of a character

Let $G$ be a finite group, let $p$ be a prime, and let $\chi\in\operatorname{Irr}(G)$. Let $\chi^0$ be the restriction of $\chi$ to the elements of $G$ of order prime to $p$, and let $\operatorname{IBr}(\chi^0)$ be the set of irreducible $p$-Brauer characters occurring in $\chi^0$. Conjecture B. One has

$$|\operatorname{IBr}(\chi^0)|\cdot\chi(1)_p\le |G|_p.$$

The inequality is known for characters of defect one, for symmetric groups, and for simple groups of Lie type in defining characteristic; its validity for other classes, including solvable groups and groups of Lie type in non-defining characteristic, remains open.