The mixed-order strong subnormalizer conjecture

Let $G$ be a finite group and let $x\in G$. Write $x=\prod_p x_p$ for the decomposition of $x$ into its $p$-parts. Assume that, for every prime $p$ dividing $o(x)$, the pair $(G,x_p)$ satisfies the strong subnormalizer conjecture. The mixed-order strong subnormalizer conjecture. There exists a bijection

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

such that $\chi(x)=\pm f(\chi)(x)$ for every $\chi\in\operatorname{Irr}^x(G)$, and, if $x_{p'}$ centralizes a Sylow $p$-subgroup of $G$, then $\chi(1)_p=f(\chi)(1)_p$ for every such $\chi$. The paper notes counterexamples to analogous unrestricted formulations and treats this statement more as a question than as an established conjecture.