Positivity of H_{P,rk} for parabolic-support posets

Prove that, for every parabolic-support poset P associated with a finite Coxeter group, the polynomial H_{P,rk}(q,t) appearing in the decomposition ∑_{n≥0} Z_{P,rk}([n+1]_q) t^n = H_{P,rk}(q,t) / ∏_{ℓ=0}^{H} (1 - q^{ℓ} t) has non-negative integer coefficients, where rk is the rank function on P and H is its maximum value.

Background

Section 5 develops a positivity criterion for the numerator H_{P,h}(q,t) of the generating series Z_{P,h}(t) = ∑{n≥0} Z{P,h}([n+1]q) t^n, showing that if a bounded, graded poset P admits an R-labelling, then H{P,rk} has non-negative coefficients.

The authors note that several classical bounded posets (e.g., noncrossing partition lattices, shard-intersection orders, and intersection lattices of hyperplane arrangements) satisfy this criterion and hence have non-negative H_{P,rk}.

They then consider parabolic-support posets associated with finite Coxeter groups: every interval in these posets is shellable and upper-semimodular (hence admits an R-labelling), but the full posets are not bounded, so the established positivity criterion does not apply. Empirically, positivity appears to hold for these posets, leading to the stated open problem.