Bi-UFS Positive Conjecture for positive semidomains

A positive semidomain is a complex semiring consisting of positive real numbers. A positive semidomain $S$ is bi-UFS if both its additive monoid $(S,+)$ and its nonzero multiplicative monoid $(S^*,\cdot)$, where $S^*:=S\setminus\{0\}$, are unique factorization monoids.

Bi-UFS Positive Conjecture. A positive semidomain $S$ is a bi-UFS if and only if

$$S=\mathbb{N}_0.$$

The semidomain $\mathbb{N}_0$ is a bi-UFS because its additive monoid is freely generated by $1$ and its multiplicative monoid is freely generated by the rational primes. The conjecture was proposed in the study of bi-atomic semirings and is one of the main motivations for the paper; its resolution is not indicated in the supplied text.