Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets

We show that if $E$ is a closed convex set in ${\mathbb{C}}^n$ $(n>1)$ contained in a closed halfspace $H$ such that $E\cap bH$ is nonempty and bounded, then the concave domain $\mathrm{\Omega }={\mathbb{C}}^n\setminus E$ contains images of proper holomorphic maps $f:X\to {\mathbb{C}}^n$ from any Stein manifold $X$ of dimension $<n$, with approximation of a given map on closed compact subsets of $X$. If in addition $2\dim X+1\le n$ then $f$ can be chosen an embedding, and if $2\dim X=n$ then it can be chosen an immersion. Under a stronger condition on $E$ we also obtain the interpolation property for such maps on closed complex subvarieties.