Sommers–Tymoczko's Weyl type subset generating-function conjecture

Let $I\subset\Phi^+$ be a lower ideal. A subset $Y\subset I$ is of Weyl type if $calpha,\beta\in Y$ and $\alpha+\beta\in I$ imply $\alpha+\beta\in Y$, and if $\gamma,\delta\in I\setminus Y$ and $\gamma+\delta\in I$, then $\gamma+\delta\in I\setminus Y$. Let $\mathcal{W}^I$ be the set of Weyl type subsets of $I$, and let $d_1^I,\dots,d_n^I$ be the dual partitions of the height distributions of positive roots in $I$. Sommers–Tymoczko's conjecture.

$$\sum_{Y\in\mathcal{W}^I}q^{|Y|}=\prod_{i=1}^n(1+q+\cdots+q^{d_i^I}).$$

This conjecture relates Weyl type subsets of a lower ideal to the dual partition of its positive-root height distribution. The source presents it as a conjecture of Sommers and Tymoczko; no resolution is supplied here.