Moretó's picky conjecture for finite groups

Let $G$ be a finite group, let $p$ be a prime, and let $x\in G$ be a picky $p$-element, meaning that $x$ lies in a unique Sylow $p$-subgroup. Let $P\in\operatorname{Syl}_p(G)$ contain $x$. Write $\operatorname{Irr}^x(G)$ for the irreducible complex characters of $G$ that are nonzero at $x$. Picky conjecture. There exists a bijection

$$f:\operatorname{Irr}^x(G)\to\operatorname{Irr}^x(\operatorname{N}_G(P))$$

such that, for every $\chi\in\operatorname{Irr}^x(G)$, $\chi(1)_p=f(\chi)(1)_p$ and $\mathbb{Q}(f(\chi)(x))=\mathbb{Q}(\chi(x))$. The conjecture proposes a local-global character correspondence; the source notes that it has been proved in several important classes, including quasisimple groups of Lie type in non-defining characteristic, symmetric groups, and $p$-solvable groups with $p\neq2$.