This is only a too long comment trying to make sense of some obscurities/omissions in Zobin's article. On page 99 there he writes $\bigcap_{\,l=1}^{\,k}W^l_\infty(\Omega)\not\subset L^\infty(\Omega)$ which suggest that he does not intend $W^l_\infty(\Omega)$ to be the usual Banach space $W^{l,+\infty}(\Omega)$ but instead a Fréchet space in whose definition $C^k(\Omega)$ indeed is intended to be the space of not necessarily bounded functions with topology of uniform convergence of derivatines on compact sets. Then his application of the Open Mapping Theorem may be the following:
For any fixed $x\in\Omega$ define $f_x$ by $\Omega\owns y\mapsto d_{\,\Omega}(x,y)\,$. Then let $U$ in the space $E=W^1_\infty(\mathbb R^n)$ be the set of all $f_1$ with first order (a.e. defined weak) partial derivatives having absolute value less than $1$ on $\mathbb R^n$. Now $U$ is an open zero neighbourhood in $E$ whence by the Open Mapping Theorem there is a zero neighbourhood $V$ in $F=W^1_\infty(\Omega)$ with $V\subseteq R\,[\,U\,]\,$. Then there is a compact set $K\subseteq\Omega$ and $\varepsilon\in\mathbb R^+$ such that for $g$ in $F$ with $|\,g\,z\,|\le\varepsilon$ for $z\in K$ and first order partials having absolute value at most $\varepsilon$ on $\Omega$ we have $g\in V$.
Fixing some $x_0\in\Omega\,$, for any $x\,,z\in\Omega$ we have $f_x\,z\le f_{x_0}\,z+f_{x_0}\,x\,$. Since $f_{x_0}$ can be extended to a continuous function on $\mathbb R^n$, there is $M_9\in\mathbb R^+$ such that $f_x\,z\le M_9$ holds for all $x\,,z\in\Omega\,$. Taking $C=\varepsilon^{-1\,}(1+M_9)\,$, for $g=C^{-1}f_x$ we see that $g\in V$ holds, and so there is $f_1\in U$ with $g=R\,f_1\,$. For $\tilde f=C\,f_1$ then $f_x\subseteq\tilde f$ with $|\,\partial^{\,\alpha}\tilde f\,z\,|< C$ for $z\in\mathbb R^n$ and $|\,\alpha\,|=1\,$.
Note that, if my interpretation above of Zobin's quite weird definition of the space $W^1_\infty(\Omega)$ as a topological linear supremum of the Fréchet space $C\,(\Omega)$ embedded in $\mathscr D'(\Omega)$ and the inverse image of the weak gradient from $L^{+\infty}(\Omega,\mathbb R^n)$ is correct, then the condition $W^1_\infty(\Omega)\subset W^1_\infty(\mathbb R^n)\,|_\Omega$ is, at least apriori, much more restrictive than it would be for the usual definition of the Sobolev space $W^{1\,,+\infty}(\Omega)$ as a Banach space.