Spectral-gap conjecture for the Berezin transform at fixed weight deficit

Let $j$ be a positive parameter, let $d\in\mathbb{N}$, and set $m=j-d$. Denote the eigenvalues of the Berezin transform $\mathcal{B}_{j,j-d}$ by $\lambda^{(0)},\lambda^{(1)},\ldots,\lambda^{(2j)}$. Spectral-gap conjecture. For every $d\in\mathbb{N}$ there exists $j_0$ such that for all $j\geq j_0$, the eigenvalues satisfy

$$\lambda^{(1)}>\lambda^{(2)},\lambda^{(3)},\ldots,\lambda^{(2j)},$$

and hence the spectral gap is

$$\gamma(\mathcal{B}_{j,j-d})=1-\lambda^{(1)}=1-\frac{(j-d)^2}{j(j+1)}=\frac{2d+1}{j}-\frac{(d+1)^2}{j(j+1)}.$$

The conjecture predicts the eventual dominance of the first nontrivial eigenvalue and gives an explicit formula for the spectral gap when the weight deficit $d$ is fixed. Its status is not resolved in the supplied source.