Saxl's arbitrary-rank symmetric p-core conjecture

For $n\in\mathbb{N}$, let ${\bf D}^{\mathbb{C}}(\lambda)$ denote the simple complex $\mathfrak{S}_n$-module labelled by the partition $\lambda$. A partition is a symmetric $p$-core if it is symmetric and has no removable rim hook of length $p$. Saxl's arbitrary-rank conjecture. For every $n\in\mathbb{N}$, there exists a symmetric $p$-core $\lambda$ for some $p\leqslant n$ such that ${\bf D}^{\mathbb{C}}(\lambda)\otimes{\bf D}^{\mathbb{C}}(\lambda)$ contains every simple $\mathbb{C}\mathfrak{S}_n$-module with positive multiplicity. This proposes a version of Saxl's tensor-square positivity conjecture for arbitrary $n$, rather than only triangular values; the supplied text gives no resolution status.