The bi-LFS/bi-UFS equivalence conjecture for positive semidomains

Let $S$ be a positive semidomain. It is a bi-LFS if both $(S,+)$ and $S^\bullet$ are length-factorial monoids (LFMs), and a bi-UFS if both $(S,+)$ and $S^\bullet$ are unique-factorization monoids (UFMs). Bi-LFS/bi-UFS equivalence conjecture. $S$ is a bi-LFS if and only if it is a bi-UFS. This conjecture would greatly restrict the possibilities for counterexamples to the question of whether $\mathbb{N}_0$ is the only positive subsemidomain of $\mathbb{R}$ that is a bi-LFS.