Strengthened Saxl conjecture for symmetric and alternating squares

Let $n=\tfrac{1}{2}k(k+1)$ and let $\rho_k=(k,\ldots,2,1)\vdash n$ be the staircase partition. Write $S^2([\rho_k])$ and $A^2([\rho_k])$ for the symmetric and alternating parts of the Kronecker square $[\rho_k]^2$. Refinement of Saxl's conjecture. The symmetric part contains every irreducible character $[\lambda]$ of $S_n$, except $[1^n]$ when $k\equiv2\pmod 4$; the alternating part contains all irreducible characters except $[n]$, $[n-1,1]$, $[n-2,2]$, additionally $[1^n]$ when $k\not\equiv2\pmod4$, and additionally $[2^3]$ when $k=3$. This strengthening is motivated by computational data and known results locating several constituents in the symmetric or alternating parts, but remains open.