Two-summand criterion for Laurent series over semidomains

Let $S$ be an additively reduced and additively atomic semidomain, and let $cmathcal{A}_{+}(S)$ denote the set of additive atoms of $S$ and $S^{\times}$ its multiplicative unit group. For a Laurent series $f$, write $\operatorname{supp}(f)$ for its support. Two-summand criterion. The following statements are equivalent:

1. $\mathcal{A}_{+}(S)=S^{\times}$.
2. Every $f\in S\llbracket x^{\pm1}\rrbracket$ with $|\operatorname{supp}(f)|>1$ can be expressed as the sum of at most two multiplicative irreducibles.
3. There exists $k\in\mathbb{N}$ such that every $f\in S\llbracket x^{\pm1}\rrbracket$ with $|\operatorname{supp}(f)|>1$ can be expressed as the sum of at most $k$ multiplicative irreducibles.

This proposed refinement characterizes exactly when a uniform finite bound on the number of multiplicative irreducible summands improves to the bound two. The source presents it as a conjectural refinement motivated by the established upper bound of three summands; no resolution is supplied.