Gromov's Uryson width conjecture for smooth domains

Let $X\subset\mathbb{R}^n$ be a smooth domain whose boundary has mean curvature satisfying

$$H_{\partial X}\geq n-1.$$

Gromov's conjecture. There should exist a continuous self-map $R:X\to X$ such that $R(X)$ has topological dimension $n-2$ and

$$\operatorname{dist}(x,R(x))\leq c_n$$

for every $x\in X$, with the best expected constant $c_n=1$. This conjecture predicts a codimension-two retraction at uniformly bounded distance and is related to upper bounds for the Uryson $1$-width of mean-convex domains. The source paper proves an upper bound for the relevant three-dimensional class, but the general conjecture stated here is not identified as resolved.