Finiteness conjecture for omega-primality of algebraic cyclic free semirings

Let $\alpha$ be a positive algebraic number such that the additive monoid $\mathbb{N}_0[\alpha]$ is atomic. For an atomic monoid $M$, write $\omega(M)$ for its omega-primality invariant, which measures the bounded number of factors needed to guarantee divisibility by a given atom.

Omega-primality finiteness conjecture. One has

$$\omega(\mathbb{N}_0[\alpha])<\infty$$

if and only if $\alpha\in\mathbb{N}$.

This conjecture proposes that finite omega-primality occurs exactly for the natural-number parameters. The source presents it after an infinite omega-primality result and related work on $\mathbb{N}_0[\alpha]$; no resolution of the general algebraic case is supplied.