A dimension growth conjecture for nondegenerate manifolds

Let $\mathcal M$ be a bounded immersed submanifold of $\mathbb{R}^{M}$ with boundary, of dimension $n$ and codimension $R$. Suppose that $\mathcal M$ is $l$-nondegenerate for $l\leq R+1$, meaning that the partial derivatives through order $l$ of a local parametrization span the ambient space almost everywhere. Let $N_{\mathcal M}(Q,0)$ denote the number of rational points of denominator at most $Q$ lying on $\mathcal M$. Dimension growth conjecture for nondegenerate manifolds. There exists a constant $c_{\mathcal M}>0$ depending only on $\mathcal M$ such that

$$N_{\mathcal M}(Q,0)\leq c_{\mathcal M}Q^{n+\epsilon},$$

for some $\epsilon>0$ and all $Q\geq1$. This is proposed as the manifold analogue of Serre's dimension growth conjecture; the source motivates it from the preceding neighborhood-counting conjecture and does not state a general proof.