The integer-coefficient supremum conjecture for maximal eigenvalue multiplicities

For $N\in\mathbb N^*$, let $\mathfrak M(N)$ be the family of admissible arrays $\mathfrak A$, and for $k\in\mathbb N^*$ let $\mathfrak m_k(N)$ be the supremum of the multiplicity of the $k$-th eigenvalue over $\mathfrak M(N)$. The entries of $\mathfrak A$ are denoted by $a_i^j$. Integer-coefficient supremum conjecture. For determining $\mathfrak m_k(N)$, it suffices to take the supremum over those $\mathfrak A$ for which $a_i^j\in\mathbb Z$. This would reduce the determination of maximal multiplicities to arrays with integer coefficients; the source presents it as an intermediate conjectural step, with no resolution supplied here.