Three-character covering conjecture for prime divisors of character degrees

Let $G$ be a finite group, and let $\rho(G)$ be the set of primes dividing the degree of some irreducible character of $G$. Write $\operatorname{Irr}(G)$ for the set of irreducible characters of $G$. Three-character covering conjecture. There exist $\chi_1,\chi_2,\chi_3\in\operatorname{Irr}(G)$ such that every prime in $\rho(G)$ divides $\chi_1(1)\chi_2(1)\chi_3(1)$. If $G$ is solvable, two characters suffice. This is proposed as a strong form of Huppert's $\rho$-$\sigma$ conjecture; its status is not specified in the source.