The harmonic-oscillator lower-bound and equality conjecture for maximal multiplicities

For $N\in\mathbb N^*$, let $\mathfrak A_N^{har}$ be the isotropic harmonic-oscillator array, let $\lambda_k^{har}$ denote its $k$-th eigenvalue, and let $m(k,\mathfrak A)$ denote the multiplicity of the $k$-th eigenvalue of an admissible array $\mathfrak A$. Write $\mathfrak m_k(N)$ for the supremum of these multiplicities over the admissible arrays. Harmonic-oscillator maximality conjecture. For every $k$ and $N$,

$$\mathfrak m_k(N)\geq m(k,\mathfrak A_N^{har}),$$

with equality for those $k$ such that $\lambda_{k-1}^{har}<\lambda_k^{har}$. The conjecture proposes the harmonic oscillator as a maximizer, at least at the indicated non-repeated eigenvalue positions. The source later states that the corresponding maximality claim is disproved by the computation $\mathfrak m_4(3)=4$, so this conjecture is refuted.