Percolative properties of the random coprime colouring

Given $u$ and $v$ in $\mathbb{Z}^d$, say that $u$ is visible from $v$ if the segment from $u$ to $v$ contains exactly two elements, which are $u$ and $v$. Take $X$ "uniformly at random in $\mathbb{Z}^d$" and colour each vertex $u$ of $\mathbb{Z}^d$ in white if $u$ is visible from $X$ and in black otherwise. Previous independent works of Pleasants-Huck and of the third author give a precise meaning to this definition. This paper is dedicated to the study of this random colouring from the point of view of percolation theory: given a reasonable graph structure on $\mathbb{Z}^d$, how many infinite black (resp. white) connected components are there?