Conjecture A on decomposition numbers of defect-two characters

Let $G$ be a finite group, let $p$ be a prime, and let $\chi\in\operatorname{Irr}(G)$ satisfy

$$|G|_p=p^2\cdot\chi(1)_p.$$

Here $n_p$ denotes the largest power of $p$ dividing the integer $n$, $\operatorname{IBr}(G)$ is a set of irreducible $p$-Brauer characters of $G$, $\chi^0$ is the restriction of $\chi$ to the elements of $G$ of order prime to $p$, and

$$\operatorname{IBr}(\chi^0)=\{\varphi\in\operatorname{IBr}(G)\mid d_{\chi\varphi}\ne0\}.$$

Conjecture A. One has $|\operatorname{IBr}(\chi^0)|\le p^2-1$ and $d_{\chi\varphi}\le p$ for every $\varphi\in\operatorname{IBr}(G)$.