Oscillation conjecture for Berezin-transform eigenvalues at fixed weight deficit

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

$$\lambda^{(0)},\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(2j)}$$

has $d$ local minima and $d$ local maxima. This predicts a precise eventual oscillation pattern in the spectrum for each fixed weight deficit, based on the numerical plots described in the source. Its status is not resolved in the supplied source.