Khintchine's conjecture for non-degenerate manifolds

Let $\mathcal{M}\subset\mathbb{R}^n$ be a $d$-dimensional non-degenerate submanifold, let $\mu_d$ be the normalised $d$-dimensional Lebesgue measure induced on $\mathcal{M}$, and let $\psi$ be an approximating function. Let $W_n(\psi)$ denote the set of simultaneously $\psi$-approximable points in $\mathbb{R}^n$. Khintchine's manifold conjecture.

$$\mu_d(\mathcal{M}\cap W_n(\psi))=\begin{cases}1,&\text{if }\displaystyle\sum_{q=1}^{\infty}\psi(q)^n=\infty,\\[2ex]0,&\text{if }\displaystyle\sum_{q=1}^{\infty}\psi(q)^n<\infty.\end{cases}$$

The source describes this as widely believed and as the expected analogue of Khintchine's theorem for non-degenerate manifolds; it presents background and partial results rather than a general proof.