Saxl's staircase tensor-square conjecture

For $k\geq 1$, let

$$\delta_k=(k,k-1,\ldots,1)$$

be the staircase partition, which has size $\binom{k+1}{2}$. For partitions of this size, $g(\delta_k,\delta_k,\mu)$ is the multiplicity of the irreducible $S_{\binom{k+1}{2}}$-module indexed by $\mu$ in the tensor square of the module indexed by $\delta_k$.

Saxl's conjecture.

$$g(\delta_k,\delta_k,\mu)>0$$

for all $k$ and all $\mu\vdash\binom{k+1}{2}$.

This is a concrete proposed choice of the symmetric partition in the tensor square conjecture. Its significance is that the staircase partition is specified uniformly rather than existentially.