Chernikov's VC-density conjecture for NIP structures

Let $(M;\ldots)$ be an NIP structure, and let $\mathcal{S}=\{S_b:b\in Y\}$ be a definable set system with $Y\subseteq M^n$. Define its VC-density by

$$\operatorname{vc}(\mathcal{S})=\limsup_{t\to\infty}\frac{\log\pi_{\mathcal{S}}(t)}{\log t}.$$

Chernikov's VC-density conjecture. The following assertions hold:

1. $\operatorname{vc}(\mathcal{S})$ is rational.
2. If $M$ is o-minimal, there is a finite set $\Delta=\Delta(n)\subseteq[0,n]\cap\mathbb{Q}$ such that $\operatorname{vc}(\mathcal{S})\in\Delta$.
3. If $M$ is o-minimal and weakly locally modular, then $\operatorname{vc}(\mathcal{S})\in\{0,1,\ldots,n\}$.

These statements formalize the proposed distinction between field-like and module-like structures through shatter-function growth. The source attributes this conjecture to Chernikov; its resolution status is not specified in the supplied text.