The undecidability conjecture for the Sigma-1 theory of Young's lattice with all constants

Let $\mathcal P$ be the set of partitions, and consider Young's lattice with a constant symbol for every partition, $\left\langle \mathcal P,\leq,\pi:\pi\in\mathcal P\right\rangle$. The all-constants Sigma-1 undecidability conjecture. The $\Sigma_1$-theory of $\left\langle \mathcal P,\leq,\pi:\pi\in\mathcal P\right\rangle$ is undecidable. This would establish undecidability at the earliest quantifier-complexity level for the language with all partition constants; the source notes that the analogous result is known for subword order, while the corresponding interpretation approach may not work directly for Young's lattice.