About an estimate of an oblique derivative problem of Laplace's equation

Question

Suppose $n\geq 2$, set $B=B_r(0)\subset \mathbb{R}^n$, $B^+=\{x\in B|x_n>0\}$, $H=\{x\in B|x_n=0\}$, Let $u\in C^\infty(B^+)\cap C^1(B^+\cup H)$ be a solution of the following oblique derivative problem: $$\begin{cases} \Delta u=0 & \text{in} & B^+,\\ au+\beta\cdot\nabla u=0 & \text{on} & H, \end{cases}$$ where the constant $a\leq 0$, $\beta$ is a constant vector with the $n$-th component $\beta_n>0$. Then $u$ can be uniquely extended to a harmoic function on $B'=B_{\lambda r}(0)$, where $\lambda=\beta_n/|\beta|$. In fact, let $v=au+\beta\cdot\nabla u$ on $B^+\cup H$, and $v(x',-x_n)=-v(x',x_n)$, then $v$ is harmonic on $B$; for any $x_0\in H$, let $$u(x_0+t\beta)=e^{-at}(u(x_0)+\int_0^t v(x_0+s\beta)e^{as}ds)$$ whenever $x_0+t\beta\in B$, then it's easy to check $u$ is smooth and harmonic in $B'$, and coincides with the previous $u$ in $B'\cap \{x_n\geq 0\}$.
The question is: Does the following estimate necessarily hold, where $C$ is a positive constant independent of $u$? $$\sup_{B'}|u|\leq C\sup_{B^+}|u|$$ I guessed it's true but could not find the answer. If it's indeed true, please give a sketch of proof, if not, I also want to see the reason.