Panova's equivalence conjecture for #P computation of Kronecker coefficients

Let $\textsc{ComputeKron}$ be the problem of computing $g(\lambda,\mu,\nu)$ from partitions $\lambda,\mu,\nu$, and let $\ell(\lambda)$ denote the number of parts of $\lambda$. Panova's conjecture. The problem $\textsc{ComputeKron}$ is in $\#\mathrm{P}$ when restricted to $\ell(\lambda)=2$ if and only if it is in $\#\mathrm{P}$ in the general case. Likewise, restricting to inputs with $\mu=\nu=(n^d)$ and $\lambda\vdash nd$ yields membership in $\#\mathrm{P}$ if and only if the general problem is in $\#\mathrm{P}$.

The claim proposes that two structured families capture the general $\#\mathrm{P}$-membership question; the source gives no resolution.