Let $\mathcal{F}$ be the Grothendieck ring of an abelian fusion category. Let $(M_i)$ be its fusion matrices and $(\mathrm{diag}(\lambda_{i,j}))$ their simultaneous diagonalization. Take $M_1=id$, so that $\lambda_{1,j}=1$. The numbers $$c_j:=\sum_i \vert \lambda_{i,j} \vert^2$$ are usually called the *formal codegrees*. For the fusion category $Rep(G)$ with $G$ finite group, by the Schur orthogonality relations, $(|G|/c_j)$ are the class sizes and $$ \sum_j \frac{1}{c_j} \lambda_{i,j} \overline{\lambda_{i',j}} = \delta_{i,i'}. $$

**Question**: Is above equality true for every abelian complex fusion category? If so, is it true for every abelian fusion ring?