Characterization of 2-distinguishable trees by branch orbit counts

Let $T$ be a tree. For a vertex $w$ and a neighbor $x$ of $w$, let $T^x$ denote the component containing $x$ after deleting $w$, let $\mu(x)$ denote the number of isomorphism types among the branches rooted at $x$, and let $a(T^x)$ denote the number of distinguishing subsets of $T^x$. Characterization conjecture. The tree $T$ is $2$-distinguishable if and only if, for any vertex $w$ and any neighbor $x$ of $w$, $\mu(x)\leq a(T^x)$ if $T^x$ is finite, and

$$\mu(x)=2^{|T^x|}$$

if $T^x$ is infinite. This would characterize 2-distinguishable trees in terms of the number of branch types and distinguishing subsets of their rooted components, extending the preceding characterization of $a$-maximum trees; the paper provides no resolution of the conjecture.