Mutation equivalence conjecture for matching field polytopes of partial flag varieties

Let $B_c^\sigma$ be the matching fields considered in the paper, with $\sigma \in S_n$ and $c \in [p-1]$. Suppose that the matching field polytope associated to $B_c^\sigma$ gives rise to a toric degeneration of $\operatorname{Gr}(k,n)$. Mutation equivalence conjecture. The matching field polytopes associated to $B_c^\sigma$ that give rise to toric degenerations of $\operatorname{Gr}(k,n)$ are all mutation equivalent for $\sigma \in S_n$ and $c \in [p-1]$. The paper proves mutation equivalence when $c=1$ and $\sigma$ belongs to a certain collection of permutations, and conjectures that the arguments extend to all matching fields $B_c^\sigma$; the claim concerns the classification of the resulting toric degenerations up to combinatorial mutation.