Spin variant of Saxl's conjecture for symmetric groups

Let $P(n)$ denote the set of partitions of $n$, and let $D^+(n)$ denote the relevant set of strict partitions indexing the spin characters under consideration. For $\lambda\in D^+(n)$, write $\langle\lambda\rangle$ for the corresponding spin character, and write $[\mu]$ for the irreducible character of $S_n$ labelled by $\mu\in P(n)$. Spin variant of Saxl's conjecture. For any $n\geq 4$, $n\neq 5$, there is a spin character $\langle\lambda\rangle$, with $\lambda\in D^+(n)$, whose square $\langle\lambda\rangle^2$ contains all $[\mu]$, with $\mu\in P(n)$. Computations with GAP motivate this conjecture; the source gives evidence from character-square calculations, while its general validity remains open.