Polynomiality conjecture for type-$A_n$ lower Bruhat interval sizes

Let $\Lambda^+$ be the dominant weight cone of type $A_n$, write a dominant weight as $\lambda=\sum_i\lambda_i\varpi_i$, and let $a\in\mathcal{F}$.

Type-$A_n$ polynomiality conjecture. The cardinality of the lower Bruhat interval is a polynomial in the coordinates $\lambda_1,\dots,\lambda_n$ of degree $n$:

$$|\mathcal{I}_a(\lambda)|\text{ is a polynomial in }\lambda_1,\dots,\lambda_n\text{ of degree }n.$$

This is presented as a weaker version of the geometric-polynomial conjecture. The paper proves a quasi-polynomial statement and notes that computations in types $\widetilde{A}_3$ and $\widetilde{B}_3$ suggest genuine polynomiality.