The tensor square conjecture for symmetric partitions

Let $g(\lambda,\lambda,\mu)$ be a Kronecker coefficient, and let $\mathbb{S}_\lambda$ denote the irreducible $S_n$-module indexed by the partition $\lambda$. Tensor square conjecture. For every $n\geq9$ there exists a symmetric partition $\lambda\vdash n$ such that

$$\mathbb{S}_\lambda\otimes\mathbb{S}_\lambda$$

contains every irreducible $S_n$-module; equivalently, $g(\lambda,\lambda,\mu)>0$ for all $\mu\vdash n$.

This generalizes Saxl's question about tensor squares and is part of the study of positivity of Kronecker coefficients. The source gives no resolution.