I am looking for a reference for the following result (if it is true):

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. Let $\Gamma_0\subset\partial\Omega$ be sufficiently regular.

Let $V_1:=\left\{\phi\in C^\infty\left(\overline{\Omega}\right):\phi\geq 0 ~\text{on}~ \Gamma_0\right\}$.

Let $V_2$ be the space of functions $u\in W^{1,p}(\Omega)$ with non-negative trace on $\Gamma_0$.

Then $V_1$ is dense in $V_2$.