Saxl's conjecture for staircase partitions

Let $\rho_m$ be the staircase partition of size $n=\binom{m+1}{2}$, and let $g(\lambda,\lambda,\mu)$ denote the Kronecker coefficient. Saxl Conjecture. For every partition $\mu\vdash n$, one has

$$g(\rho_m,\rho_m,\mu)>0.$$

This is the staircase-shape specialization of the tensor-square problem for irreducible representations of $S_n$. The source attributes it to Jan Saxl in 2012 and does not indicate a resolution.