The at-most-two-slopes conjecture for alternating surgeries

Let $K$notin $\mathcal{D}$ be a knot admitting positive alternating surgeries. Write $\mathcal{S}_{\rm alt}(K)$ for the set of alternating surgery slopes, and let $N$ denote the relevant integer parameter. At-most-two-slopes conjecture.

$$\mathcal{S}_{\rm alt}(K)\subseteq \{N-1,N\}.$$

In particular, the only knots with non-integral alternating surgeries are in $\mathcal{D}$. The theorem cited in the source verifies this for knots whose Alexander polynomials obstruct them from lying in $\mathcal{D}$; the remaining case concerns knots outside $\mathcal{D}$ sharing their Alexander polynomial with a knot in $\mathcal{D}$, and remains open.