Sharpness conjecture for Lipschitz nice dimensions

Let $(n,p)$ be a pair of dimensions, and let $\mathrm{S}^{Lip}(N,P)$ denote the set of Lipschitz stable mappings between smooth manifolds $N$ and $P$ of dimensions $n$ and $p$. Let the nice dimensions be the dimensions in which smooth stable mappings are dense.

Sharpness conjecture for Lipschitz nice dimensions. If $(n,p)$ lies in the complement of the nice dimensions or in their boundary, then $\mathrm{S}^{Lip}(N,P)$ is not dense in $C^{\infty}(N,P)$.

This asserts that the proposed Lipschitz nice dimensions are optimal: outside the nice dimensions and their boundary, Lipschitz stable mappings fail to be dense. The source presents this as an open conjecture in the context of the still widely open density problem.