Bounded VC2-dimension of quadratic Green–Sanders examples

For primes $p$ and positive integers $n$, let $\mathrm{QGS}(n,p)$ be the quadratic Green–Sanders example, and let $\mathrm{VC}_2$-dimension denote the paper's second-order VC-dimension. Quadratic Green–Sanders VC2 conjecture. There is $m\in\mathbb{N}$ such that for all $n\in\mathbb{N}$ and primes $p$, $\mathrm{QGS}(n,p)$ has $\mathrm{VC}_2$-dimension at most $m$. The conjecture contrasts with the preceding observation that large quadratic Green–Sanders examples have unbounded ordinary VC-dimension; no resolution is given.