The Hall-picky local-global conjecture

Let $G$ be a finite group, let $\pi$ be a set of primes, and let $H$ be a nilpotent Hall $\pi$-subgroup of $G$. Let $\mathcal H$ be the set of elements of $H$ that are $H$-picky, meaning that whenever $x\in H^g$, one has $g\in\mathbf{N}_G(H)$. The Hall-picky local-global conjecture. There exists a bijection

$$f:\operatorname{Irr}^{\mathcal H}(G)\longrightarrow \operatorname{Irr}^{\mathcal H}(\mathbf{N}_G(H))$$

such that it preserves $\pi$-parts of degrees, fields of character values at every $x\in\mathcal H$, and the subsets of characters nonzero at each such $x$. This proposes the appropriate Hall-subgroup analogue of the picky conjecture after the naive $\pi$-McKay counting statement fails; its status is open.