The volume conjecture for four-dimensional Liouville domains

Let $(X,\omega)$ be a four-dimensional Liouville domain such that $c_k(X,\omega)<\infty$ for all $k$.

Volume conjecture.

$$\lim_{k\to\infty}\frac{c_k(X,\omega)^2}{k}=4\operatorname{vol}(X,\omega).$$

Here $\operatorname{vol}(X,\omega)=\frac{1}{2}\int_X\omega\wedge\omega$ is the symplectic volume. The conjecture predicts that the asymptotic ECH-capacity obstruction recovers the volume obstruction for symplectic embeddings. It is verified in several examples and admits general lower-bound and inheritance results, but is not established for all four-dimensional Liouville domains.