Mutually embeddable graphs and the tree alternative conjecture

We prove that if a rayless tree T is mutually embeddable and non-isomorphic with another rayless tree, then T is mutually embeddable and non-isomorphic with infinitely many rayless trees. The proof relies on a fixed element theorem of Halin, which states that every rayless tree has either a vertex or an edge that is fixed by every self-embedding. We state a conjecture that proposes an extension of our result to all trees.