The tangledness-reduction conjecture for braid diagrams

Let $D_n$ be a disk with $n$ punctures, and let $\beta$ be a braid whose curve diagram contains a round disk with strictly positive interior tangledness. Let $\beta_+$ be a Garside generator or the inverse of a Garside generator that moves only strands inside this round disk and whose action reduces the tangledness of the diagram inside the disk.

Tangledness-reduction conjecture. The product $\beta\beta_+$ has smaller $\tau$-length than $\beta$.

This claim proposes that locally reducing tangledness by an appropriate Garside generator decreases the global length obtained from the right Garside normal form. Such a practical relationship would support algorithmic approaches to understanding braid length and the construction of efficient representatives; the supplied text does not state whether the claim has been proved or disproved.