The Bi-UF Positive Conjecture for positive semirings

A positive semiring is a subsemiring of the nonnegative cone of the real line. A complex semiring is a subset of the complex plane closed under standard addition and multiplication and containing $0$ and $1$. A semiring is bi-UFS when both its additive monoid and its multiplicative monoid with $1$ removed are unique factorization monoids. The prototypical semiring is $\mathbb{N}_0$.

Bi-UF Positive Conjecture. The prototypical semiring $\mathbb{N}_0$ is the only positive semiring that is a bi-UFS.

This conjecture asks for a classification of positive semirings whose additive and multiplicative structures both have unique factorization. The source presents it as a primary motivation and records that $\mathbb{N}_0$ is the prototypical known example, while several classes of algebraic monogenic semidomains are excluded by the results discussed.