The 0-1 law for uniformly repetitive rhombus tilings

Let $G=(V,E)$ be the adjacency graph of a uniformly repetitive (or uniformly recurrent) rhombus tiling. Let $F$ be a percolation process, monotone and freezing, on configurations ${0,1}^V$. Let $I\subset \{0,1\}^V$ be the set of invading configurations for $F$, and let $\mu$ be a Bernoulli measure. The 0-1 law conjecture.

$$\mu(I)\in\{0,1\}.$$

This conjecture proposes that invasion events for monotone freezing percolation processes on sufficiently regular rhombus tilings obey a zero-one law, despite the failure of such laws for arbitrary rhombus tilings and percolation processes. Its resolution is left open in the source.