The algebraic Weyl law

Let $(Y,A)$ be a pseudo-polarised surface, with $A$ a $\mathbb{Q}$-Cartier $\mathbb{R}$-divisor. Let $c^{\operatorname{alg}}_k(Y,A)$ denote the algebraic capacities of $(Y,A)$.

Algebraic Weyl law.

$$\lim_{k\to\infty}\frac{c^{\operatorname{alg}}_k(Y,A)^2}{k}=2A^2.$$

This is formulated as an analogue of the Weyl law for ECH capacities and is proved in many cases in the paper; the general pseudo-polarised surface case remains open.