The dimension conjecture for nondegenerate manifolds

Let $1\leq d<n$ be integers. Let $\widetilde{\mathcal M}_{n,d}$ be the class of submanifolds $\mathcal M\subset\mathbb R^n$ of dimension $d$ that are nondegenerate at every point, and let $\tau_{n,d}$ be the maximal value such that

$$\dim\mathcal S_n(\tau)\cap\mathcal M=\frac{n+1}{\tau+1}-\operatorname{codim}\mathcal M$$

whenever $1/n\leq\tau<\tau_{n,d}$ for every $\mathcal M\in\widetilde{\mathcal M}_{n,d}$. Dimension conjecture. Within this class, for $1<d<n$ one has $\tau_{n,d}=\frac{1}{n-d}$, while for curves and $n\geq2$ one has $\tau_{n,1}=\frac{3}{2n-1}$. The dimension formula is known for nondegenerate planar curves, with $\tau_{2,1}=1$, but the supplied text says that the general problem remains open for higher-dimensional curves and other subclasses of nondegenerate manifolds.