Connectivity conjecture for odd balanced pairings

A balanced pairing is a pairing of the marked points under consideration satisfying the balancedness condition defined above; two balanced pairings are connected if they are related by the allowed cut-and-glue operations and equivalences defined above. Let $n$ be the common size of two balanced pairings.

Connectivity conjecture. Any two balanced pairings of the same size $n$, where $n$ is odd, are connected.

The claim is presented as a purely combinatorial conjecture, independent of the topological input used earlier in the construction. The supplied text gives no resolution status beyond stating it as a conjecture.