Moretó's subnormaliser conjecture

Let $G$ be a finite group and let $\ell$ be a prime number. For an $\ell$-element $x\in G$, define the subnormaliser

$$\operatorname{Sub}_G(x):=\big\langle g\in G\mid \langle x\rangle\triangleleft\triangleleft\langle g,x\rangle\big\rangle.$$

Let $\operatorname{Irr}^x(G)$ denote the set of complex irreducible characters of $G$ that do not vanish at $x$. Moretó's subnormaliser conjecture. There exists a bijection

$$f_x:\operatorname{Irr}^x(G)\to\operatorname{Irr}^x(\operatorname{Sub}_G(x))$$

for which, for every $\chi\in\operatorname{Irr}^x(G)$,

$$\chi(1)_\ell=f_x(\chi)(1)_\ell$$

and

$$\mathbb{Q}(\chi(x))=\mathbb{Q}(f_x(\chi)(x)).$$

The conjecture predicts character correspondences between a finite group and the subnormalisers of its $\ell$-elements, preserving both the $\ell$-parts of character degrees and the fields generated by the corresponding character values. The source gives no resolution status for the conjecture.