Polynomial-time computation of poset dimension from a linear extension graph

Let $P$ be a poset and let $G_L(P)$ be its linear extension graph, whose vertices are the linear extensions of $P$. The dimension $\dim(P)$ is the minimum size of a realizer of $P$, that is, a set of linear extensions whose intersection is $P$.

Polynomial-time dimension conjecture. The dimension of a poset given its linear extension graph can be determined in polynomial time.

The proposition preceding this conjecture gives a quasi-polynomial-time algorithm, while the conjecture asks whether this can be improved to polynomial time. This would match the expected complexity suggested by the exponential size of a linear extension graph relative to the underlying poset.