Galois refinement of the nilpotent block conjecture

Let $p$ be a prime and let $B$ be a $p$-block of a finite group. An irreducible character in $B$ is almost $p$-rational if, writing $c$ for its conductor, $p^2$ does not divide $c$; a character has height zero when its block-theoretic height is zero. Galois refinement of the nilpotent block conjecture. The block $B$ is nilpotent if and only if all of its height zero almost $p$-rational characters have the same degree. Every $p$-block contains at least one height zero almost $p$-rational character, and the conjecture refines the known characterization using all height zero characters; it is stated as open, with extensive computations offered as motivation.