The simple-group reduction conjecture for $p$-rational characters

Let $p$ be a prime and let $S$ be a finite non-abelian simple group whose order is divisible by $p$. Simple-group reduction conjecture. The following assertions hold.

(a) Let $X$ be quasi-simple with $X/\mathbf{Z}(X)\cong S$ and $\mathbf{Z}(X)$ of order coprime to $p$. Let $Y$ be almost quasi-simple with component $F^*(Y)=X$ and $Y/X$ a $p$-group. If $Q\in\operatorname{Syl}_p(Y)$ and $\exp(Q/Q')=p^e$, then every character in $\operatorname{Irr}_{p'}(Y)$ is $\sigma_e$-invariant, equivalently has $p$-rationality level at most $e$.

(b) Let $A$ be almost simple with socle $F^*(A)=S$ and $A/S$ a $p$-group. If $\operatorname{Irr}_{p'}(B_0(S))$ contains $A$-invariant characters of $p$-rationality level $\alpha$, then it contains $A$-invariant characters of level $\beta$ for every $2\leq\beta\leq\alpha$. Here $\operatorname{Irr}_{p'}$ denotes irreducible characters of degree coprime to $p$. The conjecture is the finite-simple-group hypothesis used to reduce Hung's conjecture; the paper presents it as unresolved in general.