Greedy 0-monoid characterization of regular semi-upho functions

Let $g(x)$ be a formal power series in

$$1+x\mathbb{Z}_{\ge 0}[[x]].$$

A regular semi-upho function is the rank-generating function of a regular semi-upho poset. An infinite greedy $0$-monoid series is a sequence of greedy $0$-monoids $\{M_k^0\}_{k\ge 1}$ associated with $g(x)$.

Greedy 0-monoid conjecture. The formal power series $g(x)$ is a regular semi-upho function if and only if it admits an infinite greedy $0$-monoid series $\{M_k^0\}_{k\ge 1}$.

The forward implication would characterize all regular semi-upho functions by the recursive monoid construction, complementing the proved converse direction. The source gives no resolution status.