The Ruelle invariant conjecture for ECH capacities of generic star-shaped domains

Let $X$ be a nice star-shaped domain in ${\mathbb R}^4$. Write $e_k(X)$ for the error term in the asymptotic formula recovering the volume of $X$ from its ECH capacities, and let $\operatorname{Ru}(X)$ denote the Ruelle invariant of the contact form induced on $\partial X$ by the standard Liouville form. The Ruelle invariant conjecture. If $X$ is generic, then

$$\lim_{k\to\infty}e_k(X)=-\frac{1}{2}\operatorname{Ru}(X).$$

The conjecture predicts that the leading error in the ECH-capacity volume asymptotics is governed by the average rotation of the Reeb flow on the boundary. The paper gives a heuristic argument, but no resolution is supplied.