Sobolev homeomorphisms with gradients of low rank via laminates

Let $\Omega\subset \mathbb{R}^{n}$ be a bounded open set. Given $2\leq m\leq n$, we construct a convex function $\phi :\Omega\to \mathbb{R}$ whose gradient $f= \nabla \phi$ is a H\"older continuous homeomorphism, $f$ is the identity on $\partial \Omega$, the derivative $D f$ has rank $m-1$ a.e.\ in $\Omega$ and $D f$ is in the weak $L^{m}$ space $L^{m,w}$. The proof is based on convex integration and staircase laminates.