The exponent-realization conjecture for modular invariants and NIM-reps

Let $\cal{C}$ be a modular tensor category of rank $k$ with modular data $S$ and $T$. A modular invariant is a matrix $M$ satisfying $MS=SM$, $MT=TM$, $M_{ab}\in\mathbb{N}$ for all simple objects $a,b$, and $M_{00}=1$. Its exponent is the multiset

$$\mathcal{E}_M=\left(\mathds{1}^{M_{00}},X_1^{M_{11}},\ldots,X_{k-1}^{M_{k-1,k-1}}\right).$$

A NIM-rep $N$ associated to $\cal{C}$ has an exponent $\mathcal{E}(N)$ obtained from the multiplicities of the eigenvalues $S_{a,b}/S_{0,b}$ in the matrices $N_a$. Exponent-realization conjecture. For every rational conformal field theory described by a modular tensor category $\cal{C}$ and every modular invariant $M$, there exists a NIM-rep $N$ such that

$$\mathcal{E}(N)=\mathcal{E}_M.$$

This proposes that every modular invariant has a NIM-rep realizing the same exponent multiset. The provided text does not state whether the conjecture is known or open.