Hung's principal-block continuity conjecture for $p$-rationality

Let $G$ be a finite group and $p$ a prime. Let $B_0(G)$ denote the principal $p$-block of $G$, and write $c(\theta)_p=p^r$ for the $p$-rationality level of a character $\theta$. Hung's principal-block conjecture. If $B_0(G)$ contains an irreducible character $\chi$ of degree coprime to $p$ with $c(\chi)_p=p^\alpha$, then, for every $\beta$ with $2\leq\beta\leq\alpha$, it contains an irreducible character $\psi$ of degree coprime to $p$ with $c(\psi)_p=p^\beta$. This is the principal-block version of continuity of $p$-rationality. The paper proves it for $p=2$ and relates it to the McKay--Navarro conjecture, but the assertion for general primes remains open.