Bonato–Tardif's tree alternative conjecture

Let $G$ and $H$ be non-isomorphic graphs. We call $H$ a twin of $G$ if there are embeddings $G\to H$ and $H\to G$, that is, injective maps $V(G)\to V(H)$ and $V(H)\to V(G)$ preserving adjacency. Tree alternative conjecture. A tree has either none or infinitely many isomorphism classes of twins. This conjecture concerns the possible number of mutually embeddable but non-isomorphic graphs associated with a tree; its status is not established in the supplied source.