The undecidability and non-finite-axiomatizability conjecture for nontrivial differential posets

Let $\textbf P$ be a differential poset, meaning a graded locally finite poset with a unique least element whose up- and down-degree operators satisfy Stanley's differential-poset relation. Call $\textbf P$ nontrivial when it is not the one-element differential poset. Differential-poset theory conjecture. If $\textbf P$ is a nontrivial differential poset, then its elementary theory is undecidable and non-finitely axiomatizable. The claim proposes a uniform model-theoretic obstruction for all nontrivial differential posets; the source presents it as a possible direction and gives no resolution.