The multiplicative Dream Theorem for non-degenerate manifolds

Let $\mathcal M$ be a non-degenerate submanifold of $\mathbb R^k$, and let $\psi:\mathbb N\to[0,1)$ be non-increasing. Let $W_k^\times(\psi)$ denote the set of points in $[0,1)^k$ for which

$$\|nx_1\|\cdots\|nx_k\|<\psi(n)$$

has infinitely many solutions $n\in\mathbb N$. Multiplicative Dream Theorem. Almost no point on $\mathcal M$ lies in $W_k^\times(\psi)$ when

$$\sum_{n=1}^{\infty}\psi(n)(\log n)^{k-1}$$

converges, and almost every point on $\mathcal M$ lies in $W_k^\times(\psi)$ when the series diverges. This is presented as a more precise multiplicative analogue of the established metric theorem for non-degenerate manifolds. The corresponding dichotomy has largely eluded researchers and is open.