The reduced p-section subnormalizer conjecture

Let $G$ be a finite group, let $p$ be a prime, let $P\in\operatorname{Syl}_p(G)$, and let $x\in P$. Let $R_p(x)$ be a set of representatives for the conjugacy classes in the reduced $p$-section of $x$. The reduced $p$-section subnormalizer conjecture. There exists a bijection

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

such that it preserves $p$-parts of degrees, vanishing on every element of $R_p(x)$, fields of character values there, and the subsets of characters nonzero at each $g\in R_p(x)$. This strengthens the subnormalizer conjecture by controlling a whole reduced $p$-section; it is open in general.