Parabolic Haiman–Stanley–Stembridge h-positivity conjecture

Let $G=GL_n$, let $X$ be an invertible regular semisimple matrix, let $w\in S_n$, and let $J\subset\\{1,\ldots,n-1\\}$. The parabolic Lusztig variety $\mathcal{Y}_{w,J}(X)$ has intersection cohomology $IH^*(\mathcal{Y}_{w,J}(X))$, whose Frobenius character is denoted by $\operatorname{ch}(IH^*(\mathcal{Y}_{w,J}(X)))$. A symmetric function is $h$-positive if it is a nonnegative linear combination of complete homogeneous symmetric functions.

Parabolic h-positivity conjecture. The Frobenius character

$$\operatorname{ch}(IH^*(\mathcal{Y}_{w,J}(X)))$$

is $h$-positive for every $w\in S_n$ and every $J\subset\\{1,\ldots,n-1\\}$. This is proposed as the natural parabolic extension of the Stanley--Stembridge and Haiman conjectures; the paper notes that positivity can fail after twisting by an arbitrary local system, and the conjecture concerns the untwisted intersection cohomology.