Character degrees in principal blocks for distinct primes

Let $G$ be a finite group of order divisible by two distinct primes $p$ and $q$. We show that $G$ possesses a non-trivial irreducible character of degree not divisible by $p$ nor $q$ lying in both the principal $p$- and $q$-block whenever $G$ is one of the following: an alternating group ${\mathfrak{A}}_n$, $n\ge 4$, a symmetric group ${\mathfrak{S}}_n$, $n\ge 3$, or a finite simple classical group of type A, B, or C, defined in characteristic distinct from $p$ and $q$. This extends earlier results of Navarro-Rizo-Schaeffer Fry for $2\in \{p,q\}$, and in particular completes the proof of an instance of a conjecture of the same authors, e.g., in the case of symmetric and alternating groups.