The conjecture for the dimension threshold of nondegenerate curves

For a nondegenerate curve in $\mathbb{R}^n$, let $\tau_{n,1}$ denote the threshold governing the dimension of the set of simultaneously Diophantine-approximable points on such curves. The curve dimension-threshold conjecture. For every $n\ge2$,

$$\tau_{n,1}=\frac{3}{2n-1}.$$

The conjecture concerns the remaining upper-bound problem for curves, which is open in dimensions $n\ge3$ according to the source. Known Hausdorff-measure and dimension lower bounds provide partial progress but do not establish the asserted equality.