Two-row partition covering-number conjecture for symmetric-group characters

Let $n\geq 5$ and let $\lambda=(n-k,k)$, where $1\leq k\leq \lfloor\frac n2\rfloor$. Write $\chi_\lambda$ for the irreducible character of $S_n$ indexed by $\lambda$, let $\sigma_\lambda$ denote the corresponding character introduced in the paper, let $c(\varphi)$ denote the set of irreducible constituents of a character $\varphi$, and let $\mathrm{ccn}(\varphi;S_n)$ be its covering number. Two-row partition conjecture. The following assertions hold: if $n$ is odd, then $c(\chi_\lambda^2)=c(\sigma_\lambda^2)$ and hence $\mathrm{ccn}(\chi_\lambda;S_n)=\mathrm{ccn}(\sigma_\lambda;S_n)$; if $n$ is even and $\lambda\neq(\frac n2,\frac n2)$, then $c(\chi_\lambda^2)=c(\sigma_\lambda^2)$ for $1\leq k<\lceil\frac n4\rceil$, while otherwise $c(\chi_\lambda^3)=c(\sigma_\lambda^3)$, and in either case $\mathrm{ccn}(\chi_\lambda;S_n)=\mathrm{ccn}(\sigma_\lambda;S_n)$; finally, if $n$ is even, then

$$\lceil\log_2 n\rceil\leq\mathrm{ccn}\left(\chi_{(\frac n2,\frac n2)};S_n\right)\leq\lceil\log_2 n\rceil+1.$$

The conjecture refines observed covering-number patterns for two-row partitions and compares the irreducible characters with the associated characters $\sigma_\lambda$.