Geometric Stanley–Stembridge conjecture

For a proper smooth variety $\mathscr X$ equipped with a $G[\![z]\!]$-action and a $G[\![z]\!]$-equivariant map

$$f: \mathscr X \longrightarrow \mathrm{Gr},$$

assume that

$$f_*{\mathbb C} \cong \bigoplus_{\lambda \in \mathtt{P}^+} V^{\lambda}(\mathscr X) \text{\Am \symbol{02}} \mathsf{IC}_{\lambda} \in D_c^b(\mathrm{Gr}),$$

where $V^{\lambda}(\mathscr X) \in D_c^b(\mathrm{pt})$. Geometric Stanley–Stembridge conjecture. For every $i \in \mathbb{Z}$, the symmetric function

$$\sum_{\lambda \in \mathtt{P}^+} s_\lambda \cdot \dim H^i\bigl(V^\lambda(\mathscr X)\bigr) \in \Lambda$$

expands positively with respect to the elementary symmetric functions.

This proposes a geometric realization of the Stanley–Stembridge positivity phenomenon, asserting elementary-symmetric positivity for the graded multiplicity symmetric functions arising from the equivariant decomposition. The source does not provide evidence resolving the conjecture.