Asymptotic volume–knot Floer homology rank conjecture

Let $K$ be a knot, let $r(K)$ denote the total rank of its knot Floer homology, let $\operatorname{Vol}(K)$ denote its hyperbolic volume, and let $c$ denote the crossing number. Asymptotic volume–knot Floer homology rank conjecture. There exists a constant $a\in\mathbb{R}_{>0}$ such that the percentage of knots for which

$$\log r(K)<a\cdot\operatorname{Vol}(K)$$

converges to $1$ as $c\to\infty$. The original all-knots formulation is false because there are families with bounded volume and unbounded knot Floer homology rank; the reformulated claim concerns asymptotic density and is presented as the surviving conjecture.