The k-Schur involution conjecture

Let $\lambda$ be a $k$-bounded partition, meaning that its largest part is at most $k$. Let $\lambda^{\omega_k}$ denote its $k$-conjugate, and let $\omega$ be the usual involution on symmetric functions. The $k$-Schur involution conjecture. For every $k$-bounded partition $\lambda$,

$$\omega\,s_\lambda^{(k)}[X]=s_{\lambda^{\omega_k}}^{(k)}[X].$$

The claim generalizes the usual identity $\omega s_\lambda=s_{\lambda'}$; it is proved in the text when the maximum hook length of $\lambda$ is at most $k$, but remains unproved in general in the supplied passage.