The 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 associated to the winding statistic of the two-particle exclusion process. The covariance-matrix conjecture. If

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

then $G$ and $G'$ are isomorphic as planar leaf-rooted trees. This conjecture proposes that the winding covariance matrix recovers the underlying planar leaf-rooted tree; the paper provides evidence by distinguishing covariance matrices for two trees with the same degree sequence, but does not establish the general claim.