The picky Evseev condition

Let $G$ be a finite group, let $p$ be a prime, let $P\in\operatorname{Syl}_p(G)$, let $H=\mathbf{N}_G(P)$, and let $\mathcal P$ be the set of picky elements of $P$. Define $\mathcal C_p^{\mathcal P}(H)$ as the integer span of irreducible characters in $\operatorname{Irr}^{\mathcal P}(H)$ whose degrees are divisible by $p$. The picky Evseev condition. There exists a signed bijection

$$F:\pm\operatorname{Irr}^{\mathcal P}(G)\to\pm\operatorname{Irr}^{\mathcal P}(H)$$

such that $F(\chi)(1)_p=\chi(1)_p$ and

$$F(\chi)-\chi_H\in\mathcal C_p^{\mathcal P}(H)+I(H,P,S(G,P,H))$$

for every $\chi\in\pm\operatorname{Irr}^{\mathcal P}(G)$. This is formulated as a strengthening or analogue of Evseev's condition for the characters detected by picky elements; its general status is open in the supplied text.