  Let $n$ be a positive integer, and let $\rho_n = (n, n-1, n-2, \ldots, 1)$ be
the ``staircase'' partition of size $N = {n+1 \choose 2}$. The Saxl conjecture
asserts that every irreducible representation $S^\lambda$ of the symmetric
group $S_N$ appears as a subrepresentation of the tensor square $S^{\rho_n}
\otimes S^{\rho_n}$. In this short note we show that every irreducible
representation of $S_N$ appears in the tensor cube $S^{\rho_n} \otimes
S^{\rho_n} \otimes S^{\rho_n}$.
