 Homogeneous corner ambiguity in a quadrant Laplace problem

For tangential data with ${\beta }_1={\beta }_2=0$, the harmonic function $q=xy$ has zero prescribed data on both axes but a growing nonzero derivative. For normal data with ${\beta }_1={\beta }_2=\pi /2$, $q=\log |z|$ also has zero prescribed data, and $q_z=1/(2z)$ decays at infinity but is singular at the corner. Thus decay of the data, and even decay of the solution gradient, does not alone determine the quadrant representation. Requiring the reflected gradient to decay at infinity and satisfy $|z|H(z)\to 0$ at the corner removes both examples. Without such conditions, add homogeneous holomorphic terms obeying the reflection parity.