Refinement of the Heide–Saxl–Tiep–Zalesski conjecture for split squares

For $n\ge10$, let $P(n)$ be the set of partitions of $n$, and let $[\lambda]$ denote the irreducible character of $S_n$ indexed by $\lambda$. Refinement of the HSTZ conjecture. There exists $\lambda\in P(n)$ such that $S^2([\lambda])$ contains every irreducible character of $S_n$ except possibly $[1^n]$. There exists $\mu\in P(n)$ such that $A^2([\mu])$ contains all irreducible characters except $[n]$, $[n-1,1]$, $[n-2,2]$, and possibly $[1^n]$. Moreover, for every $n\ge10$ there is a symmetric partition $\lambda$ optimal for both split parts in this sense; and if a symmetric partition is optimal for $A^2([\lambda])$, missing only the three or four stated constituents, then $[\lambda]^2$ contains all irreducible characters. This is presented as a computationally motivated strengthening of the HSTZ conjecture and remains open.