Positive-path conjecture for Gordian adjacency of positive braid knots

Let $\hat\beta_1$ and $\hat\beta_2$ be positive braid knots. A positive path from $\hat\beta_2$ to $\hat\beta_1$ is a sequence of positive braid knots beginning at $\hat\beta_2$ and ending at $\hat\beta_1$, in which each successive knot is obtained from the preceding one by a positive crossing change. Write $\hat\beta_1\leq_g\hat\beta_2$ when $\hat\beta_1$ is Gordian adjacent to $\hat\beta_2$. Positive-path conjecture. If

$$\hat\beta_1\leq_g\hat\beta_2,$$

then there exists a positive path from $\hat\beta_2$ to $\hat\beta_1$. This would give a constructive version of Gordian adjacency within the class of positive braid knots; the supplied text does not state a proof or a resolution.