Graphical-function asymptotic expansion conjecture

Let $G$ be a graph with internal vertices $\mathcal{V}^{\mathrm{int}}$ and external vertices $\mathcal{V}^{\mathrm{ext}}=\{0,1,z\}$, such that the graphical function $f_G^{(\lambda)}$ exists. For a set of vertices $V$, let $G[V]$ be the induced subgraph containing all edges with both endpoints in $V$. For $z_0\in\{0,1\}$, let $G[V=z_0]=G/G[V]$ be the graph obtained by identifying all vertices in $V$ with $z_0$.

Graphical-function asymptotic expansion conjecture. Whenever the right-hand side exists, the asymptotic expansion at $z=z_0$ is

$$f_G^{(\lambda)}(z)=\sum_{V\subseteq\mathcal{V}^{\mathrm{int}}}f_{G[V\cup\{z_0,z\}]}^{(\lambda)}(z)f_{G[V\cup\{z_0,z\}=z_0]}^{(\lambda)}(1+O(|z-z_0|^2)),$$

and, whenever the right-hand side exists, the asymptotic expansion at $z=\infty$ is

$$f_G^{(\lambda)}(z)=\sum_{V\subseteq\mathcal{V}^{\mathrm{int}}}f_{G[V\cup\{0,1\}]}^{(\lambda)}f_{G[V\cup\{0,1\}=0]}^{(\lambda)}(z)(1+O(|z|^{-2})).$$

The result is presented as a well-tested conjecture for extracting leading asymptotic behavior of graphical functions, needed to subtract singular contributions in the iterative solution of differential equations. Its status is not resolved in the supplied text.