Topological Tree Alternative Conjecture for locally finite trees

Let $T$ be a locally finite tree, and write $S\equiv^{\sharp}T$ when $S$ and $T$ are mutually topological minors, meaning $S\leq^{\sharp}T$ and $T\leq^{\sharp}S$. Count the isomorphism classes of such trees $S$.

Topological Tree Alternative Conjecture. For a given locally finite tree $T$, the number of isomorphism classes of trees that are mutually topological minors with $T$ is either $1$ or $\mathfrak c$.

This is the topological-minor analogue of the Tree Alternative Conjecture. It gives a sharp alternative for the size of each mutual-topological-minor equivalence class, and the supplied text does not state that it has been resolved.