Regular-isotopy invariance of Cross for unknotting-number-one long knots

Let $K$ be a framed long knot of unknotting number one. Let $h$ be a homotopy which unknots $K$ and satisfies

$$ind(h)=1$$

or, respectively, $ind(h)=-1$. Let $Cross(h)$ denote the crossing-change invariant constructed from the extended Kauffman bracket. Regular-isotopy invariance conjecture. Then $Cross(h)$ is invariant under regular isotopy of $K$. For a knot of unknotting number two, the computed invariant appears to involve a standard polynomial together with Jones polynomials of knots of unknotting number one occurring in the loop. The conjecture proposes the corresponding regular-isotopy invariance in the unknotting-number-one case, but no general proof is given.