The HOMFLY slope conjecture

Let $K$ be a knot. Its exterior is $E_K=\mathbb{S}^3-N(K)$, and let $P_r(K;a,q)$ denote the $r$-colored HOMFLY polynomial. Define the set of HOMFLY slopes by

$$\mathcal{S}(K)=\left\{r^{-2}\operatorname{maxdeg}_q P_r(K;a,q)\mid r\in\mathbb{N}\right\}',$$

where the prime denotes the set of accumulation points. Let $\mathcal{B}(K)$ be the set of boundary slopes of properly embedded essential surfaces in $E_K$.

HOMFLY slope conjecture. For any knot $K$,

$$4\mathcal{S}(K)\subset \mathcal{B}(K).$$

This proposes a topological interpretation of the growth rate of the $q$-degree of the colored HOMFLY polynomial, motivated by the AJ conjecture and the slope conjecture for the colored Jones polynomial. Its status is not resolved in the supplied source.