Polynomial bound conjecture for sliding circuit sets of rigid braids

Let $B_N$ be the braid group on $N$ strands. A braid $x$ is rigid when its Garside normal form has the property that the final factor followed by the initial factor is again in normal form. Let $L$) be the Garside-length of $x$, and let $SC(x)$ denote its sliding circuit set.

Polynomial bound conjecture. There exists a constant $C$ such that, for every rigid braid $x$ with $N$ strands and Garside-length $L$,

$$|SC(x)|\leqslant C\cdot L^{N-2}.$$

Moreover, the bound holds with $C=2$ for sufficiently large $L$.

A polynomial bound on sliding circuit sets would give a polynomial bound for the classical Garside-theoretic conjugacy algorithm. The paper presents families with sliding circuit sets of order $L^{N-2}$, but the asserted universal bound remains open.