Large fully symmetric Kronecker coefficients for staircase and square partitions

Let $\rho_k$ denote the staircase partition and let $\delta_\ell$ denote the square partition. Their sizes are $|\rho_k|=\binom{k}{2}$ and $|\delta_\ell|=\ell^2$. For a self-conjugate partition $\lambda$, consider the fully symmetric Kronecker coefficient $g(\lambda,\lambda,\lambda)$. Staircase-and-square growth conjecture.

$$g(\rho_k,\rho_k,\rho_k)=\sqrt{n!}\,e^{-O(n)},\qquad n=\binom{k}{2},$$

and

$$g(\delta_\ell,\delta_\ell,\delta_\ell)=\sqrt{n!}\,e^{-O(n)},\qquad n=\ell^2.$$

These estimates would imply a stronger growth statement for fully symmetric Kronecker coefficients than the bounds proved earlier in the paper.