JCK polynomial detection conjecture for closed 3-braids

Let $\gamma$ be a braid on three strands, let $\hat{\gamma}$ be its closure, and let $\overline{\gamma}$ denote the braid obtained by taking the mirror operation. Let $\widetilde{J}$ be the Jones–Conway–Kauffman polynomial. The closed 3-braid detection conjecture. If $\hat{\gamma}$ is not isotopic to its mirror image, then

$$\widetilde{J}_{\hat{\gamma}}(a,t,z)\ne \widetilde{J}_{\hat{\overline{\gamma}}}(a,t,z).$$

This conjecture proposes that the JCK polynomial detects non-amphicheirality among closed 3-braids. The source says that the polynomial seems powerful for distinguishing closed 3-braids but supplies no proof or resolution of this assertion.