Periodic points in minimally supported surface-group graph subshifts

Let $\Sigma$ be a surface group, let $V$ be a finite set, and let $X\subset V^{\Sigma}$ be the graph subshift defined by a finite graph $\mathcal G$. A point $x\in X$ is periodic when its stabilizer has finite index in $\Sigma$. The periodic-point conjecture. If there is a shift-invariant Borel probability measure $\mu$ on $X$ such that $\Sigma$ acts minimally on the support of $\mu$, then $X$ contains a periodic point $x\in X$. Moreover, if for some $v\in V$ one has

$$\mu\bigl(\{x\in X\mid x(id)=v\}\bigr)>0,$$

then $X$ contains a periodic point $x\in X$ with $x(id)=v$. This is described as a discrete form of the preceding conjecture for hyperbolic tilings. The source gives no resolution, so the assertion remains open.