This question was addressed by Hansen and Nadirashvili in a series of papers, see, for example:
MR1315353MR1315353 Hansen, W., Nadirashvili, N., On Veech's conjecture for harmonic functionsOn Veech's conjecture for harmonic functions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 1, 137–153.
and references there.