Eventual negativity of averaged Ricci curvature in higher eigenspaces

Let $V_\ell$ denote the eigenspace indexed by $\ell$ in the Zeitlin model on $SU(N)$, and let $\widetilde{r}_\ell(N)$ be the averaged Ricci curvature in that eigenspace. Eventual negativity conjecture. For each fixed $\ell>1$, $\widetilde{r}_\ell(N)$ becomes negative for sufficiently large $N$, and

$$\widetilde{r}_\ell(N)\longrightarrow-\frac{H_\ell-1}{2}\quad\text{as }N\longrightarrow\infty,$$

where $H_\ell$ is the $\ell^{\text{th}}$ harmonic number. Numerical tables and plots exhibit this sign transition and support the stated limit, but no proof or resolution is given.