The Strong Slope Conjecture for colored Jones polynomials

Let $K\subset S^3$ be a knot. For sufficiently large $n$, write the minimum and maximum degrees of its colored Jones polynomial as quadratic quasi-polynomials, and let $js_K,js^*_K$ be the sets of quadratic coefficients and $jx_K,jx^*_K$ the corresponding sets of linear coefficients divided by $2$. A boundary slope is a slope represented by the boundary of an essential surface in the knot exterior.

Strong Slope Conjecture. Given a Jones slope $\frac{p}{q}\in js_K$, with $q>0$ and $\operatorname{gcd}(p,q)=1$, there is an essential surface $S\subset S^3\setminus K$ with $|\partial S|$ boundary components, each having slope $\frac{p}{q}$, such that

$$-\frac{\chi(S)}{|\partial S|q}\in jx_K.$$

Similarly, given $\frac{p^*}{q^*}\in js^*_K$ with $q^*>0$ and $\operatorname{gcd}(p^*,q^*)=1$, there is an essential surface $S^*\subset S^3\setminus K$ with $|\partial S^*|$ boundary components, each having slope $\frac{p^*}{q^*}$, such that

$$\frac{\chi(S^*)}{|\partial S^*|q^*}\in jx^*_K.$$

The conjecture connects the degree growth of colored Jones polynomials with boundary slopes and Euler characteristics of essential surfaces. It is known for alternating and adequate knots, several cable families, various pretzel and arborescent families, and all knots with at most nine crossings; the general case remains open.