This question was addressed by Hansen and Nadirashvili in a series of papers, see, for example:

[MR1315353](https://mathscinet.ams.org/mathscinet-getitem?mr=1315353)
Hansen, W., Nadirashvili, N.,
[On Veech's conjecture for harmonic functions](http://www.numdam.org/item?id=ASNSP_1995_4_22_1_137_0).
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 1, 137–153.

and references there.