The weak covariance-matrix conjecture for planar leaf-rooted trees

Let $G$ and $G'$ be planar leaf-rooted trees with no vertices of degree $2$. For each tree, let $\mathbf{\Sigma}_G$ and $\mathbf{\Sigma}_{G'}$ be the covariance matrices determined by the large-$t$ behavior of the winding statistic of the two-particle exclusion process on $\binom{G}{2}$ and $\binom{G'}{2}$, respectively, where $\binom{G}{2}$ is the graph of unordered pairs of distinct vertices of $G$ joined when one particle moves along an edge while the other remains fixed. The Weak Conjecture. If

$$\mathbf{\Sigma}_G=\mathbf{\Sigma}_{G'},$$

then $G$ and $G'$ have the same degree sequence. This weaker claim would still show that the winding covariance contains nontrivial information about the combinatorics of the underlying tree. The paper presents it as an open consequence that would be significant for graph configuration spaces.