Let $Q$ be a domain in the half-space $\mathbb R^n\cap\{x_n>0\}$ and part of its boundary is a domain $S$ on the hyperplane $x_n=0$. Let $u\in C(\bar Q)\cap C^2( Q)$ satisfy $\Delta u=0$ in $Q$ and for some fixed $\bar b=(b_1,\ldots,b_n)$, $b_n\ne0$, $$ \lim_{x_n\to0+}\frac{ \partial u}{\partial\bar b}=f\in C(S)\quad \hbox{a.e. on $S$}. $$ Does it follow that $\frac{ \partial u}{\partial\bar b}$ is continuous up to $S$?

A special case: if $$ \lim_{x_n\to0+}\frac{ \partial u}{\partial x_n}=0\quad \hbox{a.e. on $S$}, $$ will $u$ be smooth on $Q\!\cup\! S$? It is also enough to show that $\frac{ \partial u}{\partial x_n}$ is continuous up to $S$.

Posted earlier on MSE, but no answer.