The PBW basis parametrization conjecture for symplectic Lie algebras

Let $V(\lambda)$ be the irreducible representation of a symplectic Lie algebra with highest weight $\lambda$. Let $\mathcal{O}(P,A)_{(\lambda,0)}$ and $\mathcal{C}(P,A)_{(\lambda,0)}$ be the marked order and chain polytopes associated with the marked Hasse diagram for the symplectic Lie algebra, where the marking consists of $\lambda$ and zeroes. The integral points of $\mathcal{O}(P,A)_{(\lambda,0)}$ are the symplectic Gelfand–Tsetlin patterns of highest weight $\lambda$.

PBW basis parametrization conjecture. The lattice points in $\mathcal{C}(P,A)_{(\lambda,0)}$ parametrize a PBW basis of $V(\lambda)$ for the symplectic Lie algebras.

This conjecture connects marked chain polytopes with PBW bases and gives a polyhedral realization of representation-theoretic bases. The source states that it is proved in an article in preparation by Feigin, Fourier, and Littelmann.