Subtree-separation conjecture for the $(n-2,2)$-spectrum of trees

Let $T$ be a tree on $n$ vertices, and let $M_r^{(2)}(T)$ be its $(n-2,2)$ moments. These moments yield universal support-forest-counting combinations, and the Laplacian spectrum supplies the ordinary vertex-level spectral data. An embedded forest count means the number of copies of the corresponding forest occurring in $T$.

Subtree separation conjecture. The universal support-forest-counting combinations arising from $M_r^{(2)}(T)$, together with the Laplacian spectrum, determine all embedded forest counts of $T$. In particular they distinguish trees.

This is presented as a stronger, potentially more approachable form of the spectral support-forest-profile conjecture. It would imply that the combined spectral data reconstruct every embedded forest statistic and hence separate non-isomorphic trees.