Crew's generalized degree sequence conjecture for trees

Let $T$ be a tree. For a vertex subset $W\subseteq V(T)$, write $e(W)=|E(T[W])|$ and let $d(W)$ be the number of edges of $T$ with exactly one endpoint in $W$. The generalized degree sequence of $T$ is the multiset

$$\{(|W|,e(W),d(W)): W\subseteq V(T)\}.$$

Crew's conjecture. For any tree $T$, the generalized degree sequence of $T$ is determined by $X_T$.

This extends the fact that restricting to singleton subsets recovers the ordinary degree sequence. The source presents it as an interesting problem posed by Crew; its resolution is not indicated, so it remains open.