The t-involution conjecture for k-Schur functions

Let $\lambda$ be a $k$-bounded partition, let $\lambda^{\omega_k}$ be its $k$-conjugate, and define $\omega_t f=(\omega f)|_{t\to1/t}$. The $t$-involution conjecture. For every $k$-bounded partition $\lambda$,

$$\omega_t s_\lambda^{(k)}[X;t]=t^{-c(\lambda)}s_{\lambda^{\omega_k}}^{(k)}[X;t],$$

where $c(\lambda)$ is some nonnegative integer. The operator $\omega_t$ is shown to preserve the filtered space, while the displayed action on the $k$-Schur basis is only conjectured in the supplied text.