The braid-index–bridge-index conjecture for positive braid knots

Let $K$ be a positive braid knot, meaning a knot represented as the closure of a braid whose crossings are all positive. Let $b(K)$ denote its bridge index and $\beta(K)$ its braid index. The positive braid knot index conjecture. The braid index and bridge index of $K$ agree, that is, $\beta(K)=b(K)$. This extends the corresponding prediction for L-space knots to positive braid knots; the supplied text gives no resolution, so the conjecture remains open.