Floer simplicity detected by weak order-detected slopes

Let $M$ be a knot manifold. Write $\mathcal{S}(M)$ for the set of slopes on $\partial M$, $\mathcal{D}_{ord}^{wk}(M)$ for the set of weakly order-detected slopes, and $LO(M)$ for the space of left-orders on $\pi_1(M)$. Floer simplicity conjecture. The manifold $M$ is Floer simple if and only if

$$\mathcal{D}_{ord}^{wk}(M)\ne\mathcal{S}(M),$$

and therefore each $\mathfrak{o}\in LO(M)$ is boundary-cofinal. The conjecture is motivated by the analogous characterization in Heegaard Floer theory; the stated boundary-cofinality consequence is supported by the paper's sufficient condition involving slopes that are not weakly order-detected.