Chen–Haiman's Schur positivity conjecture for Catalan functions

Let $\Psi$ be a root ideal and let $mu$ be a partition. The Catalan function $H(\Psi;mu)$ is a symmetric function associated with this root ideal and partition. Chen–Haiman's conjecture. The Catalan function $H(\Psi;mu)$ is Schur positive, meaning that its expansion in the Schur basis has coefficients in $N$. This conjecture is presented as a conjecture of Chen and Haiman and is treated in the paper through stronger $k$-Schur-positivity results and special cases; the supplied text does not state that the full claim is resolved.