The Unipotent Wilf Conjecture for unipotent numerical monoids

Let $G$ be a unipotent linear algebraic group, and let $M=G_{\mathbb N}$. Let $S$ be a unipotent numerical monoid in $M$, with relative conductor $\mathtt{c}_M(S)$, relative sporadicity $\mathtt{n}_M(S)$, and embedding dimension $\mathtt{e}(S)$. Write $\mathtt{d}_G=\dim G$. Unipotent Wilf Conjecture. One has

$$\mathtt{d}_G\mathtt{c}_M(S)\leq \mathtt{e}(S)\mathtt{n}_M(S).$$

The conjecture generalizes the ordinary Wilf conjecture from numerical monoids to unipotent numerical monoids. It is proved in several special cases in the paper, including the fundamental monoids considered later, but remains open in general.