The $F_K$-slope conjecture

Let $F_K(x,q)$ be the $F_K$ invariant, with coefficient expansion $F_K(x,q)=x^{1/2}\sum_{n\geq 0}f_n(q)x^n$. For sufficiently large $n$, write the minimal $q$-degree as a quadratic quasi-polynomial

$$\deg_q(f_n)=a(n)n^2+b(n)n+c(n),$$

where $a(n)$, $b(n)$, and $c(n)$ are periodic functions. Define the finite set of $F_K$-slopes by

$$\fs(K)=\left\{\frac{1}{a(n)}\right\}_{n\in\mathbb N},$$

and let $\bs(K)$ be the collection of boundary slopes of essential surfaces in $S^3\setminus K$. $F_K$-slope conjecture.

$$\fs(K)\subset\bs(K).$$

This is an analogue for the $F_K$ invariant of Garoufalidis's slope conjecture for the colored Jones function. The source reports verification for the stated finite census, but the general assertion is open.