Isaacs–Navarro's conjecture on p-rationality levels

Let $p$ be a prime, let $G$ be a finite group, let $P\in\operatorname{Syl}_p(G)$, and let $\alpha\in\mathbb{Z}^{+}$. The $p$-rationality level of an irreducible character is determined by the $p$-part of its conductor. Isaacs–Navarro's conjecture. One has

$$\exp(P/P')\leq p^\alpha$$

if and only if every irreducible $p'$-degree character of $G$ has $p$-rationality level at most $\alpha$.

The source says this is confirmed for $p=2$; the if direction is confirmed for all $p$, while the only-if direction has been reduced to almost quasisimple groups. Thus the full conjecture remains open in general.