The Subnormalizer Conjecture

Let $G$ be a finite group and let $H$ be a subgroup. Define the subnormalizer

$$S_G(H)=\{g\in G\mid H\trianglelefteq\trianglelefteq\langle H,g\rangle\}$$

and the subnormalizer subgroup $\operatorname{Sub}_G(H)=\langle S_G(H)\rangle$. For a cyclic subgroup $H=\langle x\rangle$, write $\operatorname{Sub}_G(x)=\operatorname{Sub}_G(H)$. For a $p$-element $x$, define $\operatorname{Irr}^x(G)=\{\chi\in\operatorname{Irr}(G)\mid\chi(x)\neq0\}$. The Subnormalizer Conjecture. There exists a bijection

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

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

$$\chi(1)_p=f(\chi)(1)_p$$

and

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

The conjecture is motivated by computational evidence, including the symmetric-group example given in the source, and remains open.