Boyer–Gordon–Watson's surgery dichotomy conjecture for non-cable knots

Let $K$ be a knot in $S^3$. Define the sets of left-orderable and $L$-space surgery slopes by

$$\mathcal{S}_{LO}(K)=\{r\in\mathbb{Q}\mid \pi_1(K(r))\text{ is left-orderable}\},$$

and

$$\mathcal{S}_{L}(K)=\{r\in\mathbb{Q}\mid K(r)\text{ is an }L\text{-space}\}.$$

A cable of a nontrivial knot is a cable knot whose companion knot is nontrivial.

Boyer–Gordon–Watson's surgery dichotomy conjecture. If $K$ is not a cable of a nontrivial knot, then

$$\mathcal{S}_{LO}(K)\cup\mathcal{S}_{L}(K)=\mathbb{Q}$$

and

$$\mathcal{S}_{LO}(K)\cap\mathcal{S}_{L}(K)=\emptyset.$$

Thus every surgery slope belongs to exactly one of the two sets for a non-cable knot. The conjecture is motivated by the proposed L-space/left-orderability correspondence and remains open in the source.