Isaacs criterion for formal conjugacy class sizes of commutative Grothendieck rings

Let $\mathcal{F}$ be a complex commutative Grothendieck ring with simultaneously diagonalized fusion matrices having eigenvalues $(\lambda_{i,j})$. Let $(c_j)$ be the eigenvalues of $\sum_i X_iX_i^*$, with $c_1=\operatorname{FPdim}(\mathcal{F})$. Isaacs criterion. For all $i,j$,

$$\frac{\lambda_{i,j}c_1}{\lambda_{i,1}c_j}$$

is an algebraic integer. This is a conjectural strengthening of the commutative formal-codegree criterion, extending Isaacs' result and generalizing the abelian case of an open problem related to Kaplansky's sixth conjecture.