Chen–Haiman Schur-positivity conjecture for Catalan functions

Let $\Psi\subset\Delta^+$ be an upper order ideal in the positive roots and let $\mu$ be a partition. The associated Catalan function is

$$H(\Psi;\mu)(\mathbf{x};q)=\sum_\lambda K^\Psi_{\lambda\mu}(q)s_\lambda(\mathbf{x}).$$

Chen–Haiman's conjecture. The Catalan functions $H(\Psi;\mu)$ are Schur positive:

$$K^\Psi_{\lambda\mu}(q)\in\mathbb Z_{\ge 0}[q].$$

This conjecture has been resolved: these Catalan functions were identified with characters of Demazure crystals, and Schur-positive formulas were established using Kirillov–Reshetikhin crystals and rigged configurations.