Generalized Kauffman–Harary conjecture

Let $K$ be a prime alternating knot, and let $D$ be an alternating diagram of $K$ without nugatory crossings, so that $D$ is a minimal diagram. Let $X^{(2)}_K$ denote the double cover of $S^3$ branched along $K$. Generalized Kauffman–Harary conjecture. Different arcs of $D$ represent different elements of

$$H_1(X^{(2)}_K,\mathbb{Z}).$$

This conjecture concerns the relationship between minimal alternating knot diagrams and the first homology of the double branched cover. The supplied text does not state whether the conjecture is open or resolved.