The degree-monotone height-zero block bijection conjecture

Let $G$ be a finite group, let $p$ be a prime, let $B$ be a $p$-block of $G$, and let $b$ be its Brauer correspondent. Write $\operatorname{Irr}_0(B)$ for the set of height-zero irreducible characters in $B$, and similarly for $\operatorname{Irr}_0(b)$. Degree-monotone height-zero block bijection conjecture. There exists a bijection

$$\varepsilon:\operatorname{Irr}_0(B)\rightarrow\operatorname{Irr}_0(b)$$

such that $\varepsilon(\chi)(1)\leq\chi(1)$ for all $\chi\in\operatorname{Irr}_0(B)$. The conjecture extends the paper's degree-monotone bijection from the global McKay setting to arbitrary $p$-blocks. The stated result establishes related existence for blocks of maximal defect in symmetric groups, while the arbitrary-block case is left for future work.