The no-alternating-surgery conjecture for knots sharing Alexander polynomials with $\mathcal{D}$

Let $K\notin \mathcal{D}$ be a knot, and suppose that there is a knot $K'\in \mathcal{D}$ with

$$\Delta_K(x)=\Delta_{K'}(x).$$

Here $\Delta_K(x)$ denotes the Alexander polynomial of $K$, and $\mathcal{S}_{\rm alt}(K)$ denotes the set of alternating surgery slopes. No-alternating-surgery conjecture.

$$\mathcal{S}_{\rm alt}(K)=\emptyset.$$

The source states that the conjecture is needed to complete the analysis of knots outside $\mathcal{D}$ and that no examples are currently known of such knots admitting alternating surgeries.