The generalized rosette knot cylinder-knot conjecture

Let $s$ and $k$ be positive integers, let $\varepsilon_i\in\{\pm1\}$ for $1\leq i\leq s-1$, and consider the closure of the braid

$$(\sigma_1^{\varepsilon_1}\sigma_2^{\varepsilon_2}\ldots\sigma_{s-1}^{\varepsilon_{s-1}})^k.$$

A generalized rosette knot is this closed braid. Generalized rosette knot conjecture. Every generalized rosette knot is a cylinder knot. The claim proposes that all such closures arise as billiard knots in a cylinder; the supplied text does not state whether this has been proved or remains open.