Continuity for invariant characters of almost simple groups

Let $p$ be a prime, let $S$ be a finite nonabelian simple group, and let $S\trianglelefteq X\leq\operatorname{Aut}(S)$, where $X$ is almost simple and $X/S$ is a $p$-group. Let $\alpha\geq 2$. Invariant-character continuity conjecture. If $S$ has an $X$-invariant irreducible $p'$-degree character of $p$-rationality level $\alpha$, then $S$ has $X$-invariant irreducible $p'$-degree characters of every level from $2$ to $\alpha$.

The source proves this statement for $p=2$. No general proof is supplied, so the conjecture remains open for arbitrary $p$.