Bickel–Hong's contact-order conjecture for colored graph representations

Let $G$ be a graph with a single vertex colored $w$ (say vertex $j$), and let $n$ denote the length of the shortest path connecting vertex $1$ to vertex $j$. For the colored adjacency matrix $[?]$ representation

$$f(z,w)=(\mathcal{A}_G)^{-1}_{(1,1)},$$

Bickel–Hong's contact-order conjecture. The order of contact of $f$ is $2n$. This conjecture concerns the behavior of level curves near boundary singularities; it has been proved in several cases using series expansions and determinantal techniques, but its general status is not resolved in the supplied text.