Isaacs--Navarro refinement of McKay's conjecture

Let $G$ be a finite group, let $p$ be a prime, and let $P\in\operatorname{Syl}_p(G)$. Isaacs--Navarro refinement. In the situation of McKay's conjecture, there exists a bijection

$$\Omega:\operatorname{Irr}_{p'}(G)\longrightarrow\operatorname{Irr}_{p'}(N_G(P))$$

such that $\Omega(\chi)(1)\equiv\pm\chi(1)\pmod p$. This refines the numerical equality by requiring compatibility of character degrees modulo $p$. The source presents this refinement as open.