The general ECH volume conjecture for graded classes

Let $(Y,\lambda)$ be a closed connected contact $3$-manifold, let $\Gamma\in H_1(Y)$, and suppose that

$$c_1(\xi)+2\operatorname{PD}(\Gamma)\in H^2(Y;\mathbb Z)$$

is torsion. Choose an absolute $\mathbb Z$-grading on $ECH(Y,\lambda,\Gamma)$, and let $\{\sigma_k\}_{k=1,2,\ldots}$ be a sequence of elements of $ECH(Y,\lambda,\Gamma)$ with definite gradings satisfying $\lim_{k\to\infty}I(\sigma_k)=\infty$.

General ECH volume conjecture.

$$\lim_{k\to\infty}\frac{c_{\sigma_k}(Y,\lambda)^2}{I(\sigma_k)}=\operatorname{vol}(Y,\lambda).$$

Here $I(\sigma_k)$ denotes the chosen absolute grading and $c_{\sigma_k}(Y,\lambda)$ the associated spectral invariant. This generalizes the distinguished-spectrum volume conjecture and is intended to describe the asymptotics of quantitative ECH for arbitrary sequences of sufficiently high-graded classes. It remains open in the stated generality.