The positive-isotopy conjecture for scannable algebraic divides

Let $D$ and $D'$ be link-equivalent scannable algebraic divides of the same minimal index. Let $\beta(D)$ and $\beta(D')$ be their associated positive braids; two positive braids are positive-isotopic inside the solid torus when their positive braid words are related by isotopy among positive braids and cyclic shifts. The positive-isotopy conjecture. The braids $\beta(D)$ and $\beta(D')$ are positive-isotopic inside the solid torus. The preceding weaker version asserts positive isotopy without the same-minimal-index hypothesis or the qualification inside the solid torus; the displayed conjecture is the fuller version. The source gives no resolution.