Random-model counting conjecture for rational points near nondegenerate manifolds

Let $\mathcal{M}$ be a $d$-dimensional submanifold of $\mathbb{R}^n$ in Monge form, with codimension $m=n-d$, and let $N_{\mathcal{M}}(Q,\delta)$ count rational points of denominator at most $Q$ lying within the corresponding $O(\delta/q)$ neighborhood of $\mathcal{M}$. Say that $\mathcal{M}$ is $l$-nondegenerate everywhere when the partial derivatives up to order $l$ of its parametrization generate $\mathbb{R}^n$. Counting conjecture. If $\mathcal{M}$ is $l$-nondegenerate everywhere with $l=m+1$, then

$$N_{\mathcal{M}}(Q,\delta)\ll_{\mathcal{M}}\delta^mQ^{d+1}\quad\text{when }\delta\ge Q^{-\frac1m+\varepsilon}\text{ and }Q\to\infty.$$

The estimate matches the probabilistic heuristic that rational points distribute randomly near the manifold, in the range where $\delta$ is not too small. The source presents it as a plausible conjecture and notes that the behavior below the fine scale may depend strongly on the manifold.