Unimodality conjecture for atom counts in numerical power monoids

Let $H$ be a numerical monoid, and for $k,l\in\mathbb N$ let $\alpha_{k,l}(H)$ be the number of $k$-element atoms of $\mathcal P_{\mathrm{fin},0}(H)$ whose maximum is at most $l$. Unimodality Conjecture. For every $l\in\mathbb N$, there is an index $k\in\llbracket 0,l\rrbracket$ such that

$$\alpha_{0,l}(H)\le\cdots\le\alpha_{k,l}(H)$$

and

$$\alpha_{k,l}(H)\ge\cdots\ge\alpha_{l,l}(H).$$

Experiments are reported to support this unimodality assertion; the source does not state a proof or disproof.