Chambers of wiring diagrams

Given a wiring diagram (pseudo-line arrangement) of a permutation w∈Snw∈Sn, the chambers can be labeled with subsets of [n][n] and they are called chamber sets. In this short note, we show that two wiring diagrams (of same ww), can be mutated from one to another via the usual moves (coming from nil and braid relations), while freezing the chamber sets they have in common.