Prove the M-sequence conjecture for lattice Bruhat interval Kazhdan–Lusztig polynomials

Prove that, for every lattice Bruhat interval $[x,y]$ in a Coxeter group, the coefficient sequence of the Kazhdan–Lusztig polynomial $P_{x,y}(q)$ is an $M$-sequence.

Background

For lattice Bruhat intervals, the paper realizes the interval as the face lattice of a convex polytope and identifies the associated Kazhdan–Lusztig polynomial with a toric g-polynomial. Stanley’s conjecture asserts that the coefficients of every toric g-polynomial form an MM-sequence. Applying that conjectural statement to the polytopes constructed in the paper leads directly to the authors’ conjecture for lattice Bruhat intervals.

References

We conjecture that the coefficients of these Kazhdan--Lusztig polynomials form an $M$-sequence.

— Lattice Bruhat intervals are polytopal  (2610.17798 - Gao et al., 13 Oct 2026) in Conjecture 1.2, following Section 1

For each lattice Bruhat interval $[x,y]$ in a Coxeter group, give a natural construction of a commutative standard graded Artinian algebra $A_{x,y}$ with

\operatorname{Hilb}(A_{x,y};q)=P_{x,y}(q).

Can such a construction be made compatible with passage to subintervals?

— Lattice Bruhat intervals are polytopal  (2610.17798 - Gao et al., 13 Oct 2026) in Problem 1.3, Section 1