Vanishing conjecture for Kronecker coefficients of square partitions

For $m\ge 7$, let $(m^m)$ be the square partition of $m^2$, let $\mu'$ denote the conjugate partition of $\mu$, and define

$$S=\{(m^2-3,2,1),(m^2-4,3,1),(m^2-j,1^j)\mid j\in\{1,2,4,6\}\}.$$

Write $g(\lambda,\lambda,\mu)$ for the Kronecker coefficient. Square-partition vanishing conjecture.

$$g((m^m),(m^m),\mu)=0$$

if and only if $\mu\in S$ or $\mu'\in S$. The paper proves the forward direction and reports partial progress toward the converse, including positivity for three-row partitions and near-hooks.