Tensor-square conjecture for symmetric groups

For each positive integer $n$, let $V_\lambda$ denote the irreducible representation of $S_n$ indexed by a partition $\lambda$ of $n$. Tensor-square conjecture. There exists a partition $\lambda$ of $n$ such that

$$V_\lambda\otimes V_\lambda$$

contains all irreducible representations of $S_n$. The conjecture asks for a single suitable partition for every $n$ and is presented as an additional open problem alongside the Saxl conjecture.