The conductor–sporadicity ratio conjecture for unipotent numerical monoids

Let $n\geq 2$, let $M=\mathbf P(n,\mathbb N)$, and let $S$ be a unipotent numerical monoid in $M$. Let $\varphi(S)$ be its image in $\mathbb N^{n-1}$ under the monoid isomorphism $\varphi$, and let $c(\varphi(S))$ and $n(\varphi(S))$ denote the generalized conductor and sporadicity defined from its hole set. Conductor–sporadicity ratio conjecture. One has

$$\frac{\mathtt{c}_M(S)}{c(\varphi(S))}\leq\frac{\mathtt{n}_M(S)}{n(\varphi(S))}.$$

The paper proposes this inequality independently of the relationship between the Unipotent Wilf and Generalized Wilf conjectures. No proof or disproof is supplied, so its general validity remains open.