The dimension threshold conjecture for nondegenerate manifolds of dimension greater than one

Let $\tau_{n,d}$ denote the threshold governing the dimension of the set of simultaneously Diophantine-approximable points on nondegenerate $d$-dimensional manifolds in $\mathbb{R}^n$. For $1<d<n$, the dimension threshold conjecture asserts that

$$\tau_{n,d}=\frac{1}{n-d}.$$

This value is motivated by the hard upper bound arising from nondegenerate manifolds containing rational subspaces. The conjecture remains open within the class of nondegenerate manifolds under consideration.