Lipschitz nice-dimensions conjecture

Let $N$ be a compact smooth manifold of dimension $n$ and $P$ a smooth manifold of dimension $p$. Define the Lipschitz nice dimensions to be the pairs $(n,p)$ for which the set $\mathcal{S}^{Lip}(N,P)$ of Lipschitz stable mappings is dense in $C_{pr}^{\infty}(N^{n},P^{p})$. Mather's nice dimensions are the pairs for which smooth stable mappings are dense.

Lipschitz nice-dimensions conjecture. The Lipschitz nice dimensions contain Mather's nice dimensions and their boundary.

This conjecture concerns the still widely open problem of density of Lipschitz stable mappings. The stated result is intended to identify the full range of dimensions in which Lipschitz stable mappings are dense.