M. Shapiro's plabic-graph mutation conjecture

Let $P$ and $P'$ be plabic graphs, and associate to each its quiver. Two plabic graphs are move equivalent when they are related by local moves and changes of the colors of boundary vertices; their quivers are mutation equivalent when one can be transformed into the other by iterated quiver mutations. M. Shapiro's conjecture. Two plabic graphs are move equivalent if and only if their associated quivers are mutation equivalent. The conjecture is stated as the converse to the paper's proposition that the indicated local moves preserve mutation equivalence of the associated quivers; the supplied source gives no resolution.