Divergence from two non-isomorphic minimal convergent subgraphs

Let $G$ be a connected graph. The notation $\Lambda_n$ denotes the class of graphs whose $P_n$-line graph sequences converge, and $G_1,G_2$ are subgraphs of $G$.

Non-isomorphic subgraph conjecture. If $G_1\not\cong G_2$ and $G_1,G_2\in\Lambda_n$, then $G$ has a sequence that diverges by order.

The preceding theorem establishes divergence when the two distinct subgraphs belong to the more restricted family $\delta_n$; this conjecture proposes the corresponding statement for arbitrary non-isomorphic subgraphs in $\Lambda_n$.