Dynnikov–Wiest conjecture on linear-length sigma-consistent braid representatives

Let $B_n$ be the braid group with generating set $\Sigma=\{\sigma_1,\ldots,\sigma_{n-1}\}$, and let $l_\Sigma(w)$ denote the $\Sigma$-length of a braid $w$. A braid word is $\sigma$-consistent if it is trivial, $\sigma$-positive, or $\sigma$-negative, where these notions are defined using the standard generators and their inverses. Dynnikov–Wiest conjecture. There are numbers $\lambda,c$ such that every braid $w\in B_n$ has a $\sigma$-consistent representative whose $\Sigma$-length is bounded linearly by the $\Sigma$-length of $w$; that is, the representative has length at most $\lambda l_\Sigma(w)+c$. The conjecture concerns whether the currently known exponential bound for such representatives can be replaced by a linear one. Its resolution is not supplied in the paper.