Navarro's principal-block characterization of the continuity gap

Let $G$ be a finite group, let $p$ be a prime, and let $P\in\operatorname{Syl}_p(G)$. Define

$$K=\operatorname{N}_G(P)/\operatorname{O}_{p'}(\operatorname{N}_G(P))\Phi(P),$$

and let $Q\in\operatorname{Syl}_p(K)$. Call a character almost $p$-rational when it has degree coprime to $p$ and $p$-rationality level at most $1$. Navarro's principal-block characterization. The following are equivalent: (i) every almost $p$-rational character in $\operatorname{Irr}_{p'}(B_0(G))$ is $p$-rational; (ii) for every $1\neq y\in Q$,

$$\operatorname{N}_K(\langle y\rangle)/\operatorname{C}_K(y)\cong\operatorname{Aut}(\langle y\rangle),$$

and $\operatorname{N}_K(\langle y\rangle)/Q$ acts trivially on the set of conjugacy classes of $\operatorname{C}_K(y)/Q$. This is the principal-block analogue of the continuity-gap characterization. The paper states it as a conjecture and leaves the general assertion open.