The prime-graph complement characterization for strictly T-solvable groups

Let $T$ be a nonabelian finite simple group and let $k$ be a prime number. For each $\chi\in\operatorname{IBr}_k(T)$, define $A_{\chi}$ to be the set of edges

$$\{k-q \mid \textnormal{there \exists a } g\in T \textnormal{ such that } \operatorname{o}(g) = q \textnormal{ and } \frac{1}{\operatorname{o}(g)}\sum_{x \in \langle g \rangle}\chi(x) > 0\}.$$

Prime-graph complement characterization. For every strictly $T$-solvable group $G$, there exists a set $Y\subseteq\bigcup_{p\in\pi(T)}\operatorname{IBr}_p(T)$ such that

$$\operatorname{\overline{\Gamma}}(G)[\pi(T)] = \operatorname{\overline{\Gamma}}(T)\setminus\left(\bigcup_{\chi\in Y}A_\chi\right).$$

This proposes a uniform description of the induced prime-graph complement on the set of prime divisors of $T$ for strictly $T$-solvable groups, in terms of the prime-graph complement of $T$ and the edge sets arising from Brauer characters. The source presents it as a direction for future work; no resolution is supplied.