Internal commutativity conjecture for sliding circuit sets

For $N\geqslant 5$, let $B_N$ be the braid group on $N$ strands. Let $SC(x)$ be the sliding circuit set of a braid $x$, and call a braid maximising when it maximises $|SC(x)|$ among all rigid pseudo-Anosov braids in $B_N$ of a fixed Garside-length $L$. An ouroboros is a subsurface of the punctured disk that is almost invariant under the braid, and $SC(x)$ is obtained from $x$ by sliding ouroboroi when every element arises through such slithering operations.

Internal commutativity conjecture. For every $N\geqslant 5$, there exists an integer $L_N$ such that, whenever $L\geqslant L_N$ and $x\in B_N$ is maximising among rigid pseudo-Anosov braids of length $L$, $x$ has $N-2$ ouroboroi and $SC(x)$ is obtained from $x$ by sliding ouroboroi.

This is the paper's principal structural conjecture: large sliding circuit sets should reflect internal commutativity arising from geometric subsurfaces. It remains open; the authors report that their counterexamples without ouroboroi occur only at short lengths.