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 $V_2$.

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

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

Then $V_1$ is dense in $V_2$.

If we consider the case of "$\phi = 0$" instead of "$\phi \geq 0$", of course an analogous result for $\Gamma_0=\partial\Omega$ is well known, and there is a result that it stays true if $\Gamma_0\subset\partial\Omega$ is relatively open and has relative Lipschitz boundary.

It would already be very helpful to have a reference for the result with $\phi \geq 0$ and $\Gamma_0=\partial\Omega$.