Bounded sliding circuit sets for powers of rigid elements

Let $G$ be a Garside group and let $x\in G$ be rigid. For each positive integer $n$, let $SC(x^n)$ denote the sliding circuits set of $x^n$. Boundedness conjecture. For any rigid element $x$, the sequence

$$\left( |SC(x^n)| \right)_{n\in\mathbb N}$$

is bounded. This would clarify how sliding circuit sets behave under powers and could contribute to efficient conjugacy algorithms; the statement is known for rigid pseudo-Anosov elements in braid groups, but is open for general rigid elements.