Equality of regular and weak order-detected slopes

Let $M$ be a knot manifold. The sets of regular order-detected and weakly order-detected slopes on its boundary are denoted by $4\mathcal{D}_{ord}(M)4$ and $4\mathcal{D}_{ord}^{wk}(M)4$, respectively, as subsets of the slope set $4\mathcal{S}(M)4$. Equality conjecture. If $M$ is a knot manifold, then

$$\mathcal{D}_{ord}(M)=\mathcal{D}_{ord}^{wk}(M).$$

Weak order-detection records boundary data from left-orders, while regular order-detection additionally incorporates conjugacy invariance; the conjecture asserts that these notions nevertheless coincide for every knot manifold.