The abelian-Sylow strong subnormalizer conjecture

Let $G$ be a finite group with abelian Sylow $p$-subgroups. Let $P\in\operatorname{Syl}_p(G)$ and let $x\in P$ be picky. Let $S_p(x)$ be a set of representatives for the conjugacy classes in the $p$-section of $x$. The abelian-Sylow strong subnormalizer conjecture. There exists a bijection

$$f:\operatorname{Irr}^{S_p(x)}(G)\longrightarrow \operatorname{Irr}^{S_p(x)}(\operatorname{Sub}_G(x))$$

such that it preserves $p$-parts of degrees, satisfies $\chi(g)=\pm f(\chi)(g)$ for every $g\in S_p(x)$, and restricts to bijections between the characters nonzero at each such $g$. The source says this is expected when Sylow subgroups are abelian and does not establish it in general.