Radial behavior of positive harmonic Bloch functions

Let $u$ be a strictly positive harmonic Bloch function on the upper half-space ${\mathbb R}_+^{d+1}$. Then the set
$$ \left\{x\in{\mathbb R}^d:\ \limsup_{y\to 0+}|{\log u(x, y)}|<\infty\right\} $$
has the maximal Hausdorff dimension.