Let $\Gamma$ be a finitely generated group and $\mu$ a symmetric measure of finite support on $\Gamma$. Let $\partial_{M}\Gamma$ be the Martin boundary of $(\Gamma,\mu)$ and let $\partial^{min}_{M}\Gamma$ be the subset consisting of positive harmonic functions.

My question is: what are some conditions on the group or measure that guarantee $\partial^{min}_{M}\Gamma$ is a closed subset of $\partial_{M}\Gamma$?

There are many examples (e.g. hyperbolic groups) where all points in the Martin boundary are minimal, but otherwise examples are hard to come by.

Is there an explicit example where $\partial^{min}_{M}\Gamma$ is not closed in $\partial_{M}\Gamma$?