Yes, it does. This is sometimes called *Kuran's theorem*, proved by Ülkü Kuran in 1972, see [1], after preliminary works by a number of other researchers.

Reference:

[1] Ülkü Kuran, *On the mean-value property of harmonic functions*. Bull. London Math. Soc. 4 (1972): 311–312, [DOI:10.1112/blms/4.3.311](https://doi.org/10.1112/blms/4.3.311).