The NP-completeness conjecture for the Sigma-1 theory of Young's lattice

Let $\textbf{Y}=\left\langle \mathcal P,\leq\right\rangle$ be Young's lattice, where $\mathcal P$ is the set of partitions ordered by inclusion of their Young diagrams. The Sigma-1 theory conjecture. The $\Sigma_1$-theory of Young's lattice $\textbf{Y}=\left\langle \mathcal P,\leq\right\rangle$ is $\textbf{NP}$-complete. The preceding discussion establishes that this theory is at least $\textbf{NP}$-hard if it is decidable; the conjecture asks for the matching upper bound. An affirmative answer would give an $\textbf{NP}$-characterization of the posets that embed in Young's lattice.