Four-character Fitting-index conjecture

Let $G$ be a finite group, and let $\mathbf F(G)$ denote its Fitting subgroup. Write $\operatorname{Irr}(G)$ for the set of irreducible characters of $G$. Four-character Fitting-index conjecture. There exist $\chi_1,\chi_2,\chi_3,\chi_4\in\operatorname{Irr}(G)$ such that

$$|G:\mathbf F(G)|\mid\chi_1(1)\chi_2(1)\chi_3(1)\chi_4(1).$$

If $G$ is solvable, two characters suffice. This is proposed as a strong form of the generalization of Gluck's conjecture; its status is not specified in the source.