ACCP conjecture for cyclic Puiseux monoids

Let $r\in\mathbb{Q}_{>0}$, and let $S_r^\bullet$ denote the cyclic Puiseux monoid associated to $r$. Write $\mathsf{n}(r)$ and $\mathsf{d}(r)$ for the numerator and denominator of $r$ in lowest terms. A commutative monoid satisfies the ascending chain condition on principal ideals (ACCP) if every ascending chain of principal ideals eventually stabilizes.

ACCP conjecture. If

$$r\in\mathbb{Q}_{>0},\qquad \mathsf{n}(r)>1,\qquad \mathsf{d}(r)>1,$$

then $(S_r^\bullet,\cdot)$ satisfies the ACCP.

This conjecture would imply that $(S_r^\bullet,\cdot)$ is atomic in the setting discussed immediately before the conjecture, even when $\mathsf{d}(r)$ is not prime. The supplied text does not indicate whether the conjecture has been resolved.