The embedding conjecture for infinite prime graphs

A prime graph is a graph with no nontrivial module, and an infinite prime graph is a prime graph with infinitely many vertices. Let $G_\mu$ denote the graph associated with a $0$-$1$ sequence $\mu$ on an interval of $\mathbb{Z}$, and let the finitely many graphs displayed in Figures 1 and 2 of the source be the listed finite exceptional graphs. The embedding conjecture. Every infinite prime graph contains, as an induced subgraph, one of those displayed graphs or a graph $G_\mu$ for some $0$-$1$ sequence $\mu$ on an interval of $\mathbb{Z}$. The conjecture proposes a complete list of unavoidable induced subgraphs for infinite prime graphs; the source gives no resolution of it.