Novak–Rhoades numerical Kronecker-coefficient conjecture

For $n\geq3$, let $\nu\vdash n$, let $g_{\lambda\mu}^{\nu}$ be the Kronecker coefficient in $V^\lambda\otimes V^\mu$, and let $H_\lambda$ denote the hook product of the partition $\lambda$. Novak–Rhoades' numerical conjecture. For any $n\geq3$ and $\nu\vdash n$,

$$\sum_{\substack{\lambda\vdash n\ell(\lambda)=k-1}}\sum_{\substack{\mu\vdash n\ell(\mu)=k+1}}\frac{g_{\lambda\mu}^{\nu}}{H_\lambda H_\mu} \leq \sum_{\substack{\lambda\vdash n\ell(\lambda)=k}}\sum_{\substack{\mu\vdash n\ell(\mu)=k}}\frac{g_{\lambda\mu}^{\nu}}{H_\lambda H_\mu}$$

for all $2\leq k\leq n-1$. This is stated as an equivalent numerical formulation of the equivariant conjecture, and summing over $\nu$ recovers the earlier hook-length inequality and hence Chen's log-concavity conjecture. The source presents it as open.