Bonato–Tardif rooted tree alternative conjecture

A rooted tree $(T,r)$ is a tree $T$ with a distinguished vertex $r$. Two rooted trees $(T,r)$ and $(T',r')$ are twins, or mutually embeddable, when there are injective graph homomorphisms $\phi \colon T \rightarrow T'$ and $\psi \colon T' \rightarrow T$ satisfying $\phi(r)=r'$ and $\psi(r')=r$. Let $m(T,r)$ be the number of isomorphism classes of rooted trees $(T',r')$ mutually embeddable with $(T,r)$. Rooted tree alternative conjecture. Every rooted tree $(T,r)$ has twin number either $1$ or $\infty$.

The source says that this was implicitly conjectured by Bonato and Tardif. It is the rooted analogue of the tree alternative conjecture and is proved in the paper.