Asymptotic square-root deviation conjecture for knot slope and signature

Let $K$ be a hyperbolic knot in $S^3$. Write $\sigma(K)$ for its signature, $\operatorname{slope}(K)$ for its slope invariant, and $\operatorname{vol}(K)$ for its hyperbolic volume. A property holds asymptotically almost surely if its probability for knots with $n$ crossings tends to $1$ as $n\to\infty$.

Slope-signature deviation conjecture. There are constants $b$ and $c$ such that, for any hyperbolic knot $K$ in $S^3$,

$$|2\sigma(K)-\operatorname{slope}(K)|\leq b\sqrt{\operatorname{vol}(K)}+c$$

asymptotically almost surely.

This proposes a more precise asymptotic relationship between the slope and the signature. It is motivated by the observed smallness of the normalized quantity involving their difference, but no resolution is supplied here.