The alternating-surgery tangle-replacement conjecture

Let $\mathcal{A}$ be the set of knots admitting a positive alternating surgery, and let $\mathcal{D}$ be the set of knots in $\mathcal{L}$ that are branched double-covers of an unknotting arc in an alternating diagram, where $\mathcal{L}$ is the set of knots admitting a positive $L$-space surgery. An alternating surgery is a surgery whose resulting $3$-manifold is the double branched cover of an alternating knot or link.

Alternating-surgery tangle-replacement conjecture. Every alternating surgery arises as tangle replacement on an almost-alternating diagram of the unknot; equivalently,

$$\mathcal{A}=\mathcal{D}.$$

The conjecture would characterize all knots admitting alternating surgeries via the construction described in the paper. The paper notes that all known examples lie in $\mathcal{D}$ and that every known non-integer alternating surgery has a representative in $\mathcal{D}$, while the integer-surgery case remains unresolved.