  We prove a conjecture of Geiss, Leclerc and Schr\"{o}er, producing cluster
algebra structures on multi-homogeneous coordinate ring of partial flag
varieties, for the case $G_2$. As a consequence we sharpen the known fact that
coordinate ring of the double Bruhat cell $G^{e,w_0}$ is an upper cluster
algebra, by proving that it is a cluster algebra.
