Generalised Saxl conjecture for staircase-like partitions

Let $n$ be a positive integer, and let $\lambda\vdash n$ be a staircase-like partition, meaning a self-conjugate partition satisfying one of the three conditions described in the source: it lies between consecutive staircase partitions in the applicable parity cases, or between the indicated wider staircase bounds when no partition lies between consecutive staircases. Write $[\lambda]$ for the corresponding irreducible representation of $S_n$. Generalised Saxl conjecture. If $n\notin\{2,4,9\}$, then $[\lambda]\otimes[\lambda]$ contains every irreducible representation of $S_n$ as a constituent. This extends Saxl's conjecture from staircase partitions to the staircase-like partitions defined in the paper. The supplied text gives no resolution or partial-status evidence for this generalisation.