Regularity of quasilinear PDE up to a smooth part of the boundary

Question

Let $\Omega \subset \mathbb{R}^N$, $N \geq 2$, be a bounded domain. Consider the Dirichlet problem $$ -\Delta_p u = |u|^{q-2} u \text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega, $$ where $\Delta_p u$ is the standard $p$-Laplacian and $1

It is known from DiBenedetto that any (weak) solution $u \in W_0^{1,p}(\Omega)$ of the above problem belongs to $C^{1,\beta}(\Omega)$.

If $\partial\Omega$ is of class $C^{1,\alpha}$ for some $\alpha \in (0,1)$, then it is known from Liebermann that any solution $u$ of the above problem belongs to $C^{1,\beta}(\overline{\Omega})$, i.e., up to the boundary.

Suppose now that $\partial \Omega$ is only piecewise $C^{1,\alpha}$-smooth. (Under "piecewise" I mean something relatively good, as in the case of a cube.)

Is it true that any solution $u$ will be $C^{1,\beta}$-smooth up to a $C^{1,\alpha}$-smooth piece of the boundary?

For the case $p=2$ the similar results have to be known, e.g., from Grisvard. However, I cannot find any explicit reference for the general nonlinear case.

I would appreciate any references or hints!