Giannelli's degree-preserving refinement of the McKay conjecture

Let $G$ be a finite group, let $p$ be a prime, and let $P\in\operatorname{Syl}_p(G)$. Write $\operatorname{Irr}_{p'}(G)$ for the irreducible complex characters of $G$ whose degrees are not divisible by $p$. Giannelli's refinement. There should be a bijection

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

such that $\chi^*(1)\leq\chi(1)$ for every $\chi\in\operatorname{Irr}_{p'}(G)$. This refinement would imply the sum-of-squares inequality by comparing corresponding character degrees. The source presents it as a recent conjecture and does not state a general resolution.