Length-set realization conjecture for non-torsion power monoids

Let $H$ be a monoid that is not torsion, and let $\mathcal P_{\mathrm{fin},1}(H)$ be its finitary power monoid containing the identity. A length set is the set of factorization lengths of an element. Length-set realization conjecture. Every non-empty subset of the integers greater than one is the length set of some $X\in\mathcal P_{\mathrm{fin},1}(H)$. This would describe a particularly unrestricted system of length sets for non-torsion power monoids; the source presents it among open problems and gives no resolution.