Cubing conjecture for large sliding circuit set graphs

Let $SC(x)$ be the sliding circuit set of a braid $x$, and form its sliding circuit set graph by taking the elements of $SC(x)$ as vertices and minimal conjugations as edges.

Cubing conjecture. If the sliding circuit set is large, then large regions of the sliding circuit set graph should be cubical.

This conjecture predicts that the commuting conjugations produced by slithering ouroboroi assemble into large cubes, although branching can also occur. It is a qualitative open problem about the global geometry of sliding circuit set graphs.