The integer-valued Iitaka dimension conjecture for numerical cycle classes

Let $X$ be a projective variety over $\mathbb{C}$, let $N_k(X)_{\mathbb{Z}}$ denote the group of $k$-cycle classes up to numerical equivalence, and let $\alpha \in N_k(X)_{\mathbb{Z}}$. If some positive multiple of $\alpha$ is represented by an effective cycle, define its Iitaka dimension by

$$\kappa(\alpha):= (n-k) \sup \left\{ r \in \mathbb{R}_{\geq 0} \mid \limsup_{m \to \infty} \frac{\operatorname{mc}(m\alpha)}{m^r} > 0 \right\},$$

where $n=\dim X$ and $\operatorname{mc}(\alpha)$ is the largest number of general points contained in an effective cycle of class $\alpha$; otherwise set $\kappa(\alpha)=-\infty$. Integer-valued Iitaka dimension conjecture.

$$\kappa(\alpha) \in \mathbb{Z}_{\geq 0} \cup \{-\infty\}.$$

The conjecture predicts that the asymptotic growth invariant of a numerical cycle class takes only integral values, analogous to the classical Iitaka dimension of divisors. The paper develops evidence for this prediction, but does not establish it in general.