Saxl's conjecture on tensor squares of staircase representations

Let $k$ be a positive integer, set $n=k(k+1)/2$, and let $\lambda=(k,k-1,\ldots,2,1)$ be the staircase partition of $n$. Let $\sigma_\lambda$ be the irreducible representation of $S_n$ indexed by $\lambda$. Saxl's conjecture. The tensor product $\sigma_\lambda\otimes\sigma_\lambda$ contains every irreducible representation of $S_n$ as a constituent. This is a well-known open problem about decomposing tensor products of irreducible representations of symmetric groups; the paper presents progress and generalisations, but the conjecture itself remains open.