Huang's asymptotic conjecture for rational points near curved manifolds

Let $\mathcal{M}$ be a bounded immersed submanifold of $\mathbb{R}^M$ with boundary, and let $R=M-\dim \mathcal{M}$. Suppose that $\mathcal{M}$ satisfies proper curvature conditions. Then there is a constant $c_{\mathcal{M}}>0$ depending only on $\mathcal{M}$ such that

$$N(\mathcal{M};Q,\delta)\sim c_{\mathcal{M}}\delta^R Q^{\dim \mathcal{M}+1}$$

when $\delta\geqslant Q^{-1/R+\epsilon}$ for some $\epsilon>0$ and $Q\to\infty$.

Huang's conjecture. The asymptotic above should hold in the stated range of $\delta$.

This conjecture predicts the probabilistic main term for rational points close to manifolds under suitable curvature hypotheses, extending known lower-bound results and describing the threshold at which the expected density becomes visible. The supplied text gives no resolution status.