The move-connectivity conjecture for rotationally symmetric plabic graphs

Let $G$ and $G'$ be minimal rotationally symmetric plabic graphs, and let $f_G$ and $f_{G'}$ denote their associated permutations.

Move-connectivity conjecture. If

$$f_G=f_{G'},$$

then $G$ and $G'$ are related through a sequence of rotationally symmetric moves.

Each rotationally symmetric move preserves the associated permutation, so the conjecture asks whether all minimal rotationally symmetric representatives of a given permutation lie in one move-equivalence class. This is presented as an expected extension of the preceding lemma and is not proved in the supplied text.