The maximum VC-dimension conjecture for bounded-index power subgroups

Let $r$ be a positive integer of bounded index, meaning $r\ll 1$, and let

$$\Gamma^{(r)}=\{x^r:x\in\mathbb{F}_q^\times\}\subseteq\mathbb{F}_q^\times$$

be the subgroup of $r$th powers in the finite field $\mathbb{F}_q$. Define

$$\alpha_q^{(r)}=\frac{\operatorname{VCdim}_{\mathbb{F}_q}(\Gamma^{(r)})}{\log_2q},\qquad \underline{\alpha}^{(r)}=\liminf_{q\to\infty}\alpha_q^{(r)}.$$

Maximum VC-dimension conjecture for power subgroups. One has

$$\underline{\alpha}^{(r)}=1.$$

Consequently, $\operatorname{VCdim}_{\mathbb{F}_q}(\Gamma^{(r)})=(1+o_r(1))\log_2q$ as $q\to\infty$. This is stated as the bounded-index generalization of the quadratic-residue expectation; the supplied text says that analogous partial progress is proved, but does not give a resolution of the conjecture.