The polynomial bound conjecture for cyclic slidings

Let $B_n$ be the braid group on $n$ strands, let $x\in B_n$ have canonical length $\ell$, and let $\mathfrak s$ denote cyclic sliding. Define $t$ to be the minimal positive integer such that

$$\mathfrak s^k(x)=\mathfrak s^t(x)$$

for some $k$ with $0\leqslant k<t$.

Polynomial bound conjecture. The integer $t$ is bounded by a polynomial in $\ell$ and $n$.

Such a bound would imply polynomial complexity for the algorithm finding elements in stabilized sets of sliding circuits, provided the relevant parameter is polynomially bounded. The source calls this a well-known conjecture and gives no resolution, so its status is open.