Spectral support-forest-profile conjecture for trees

Let $T_1$ and $T_2$ be trees on $n$ vertices. The $(n-2,2)$ support-forest profile is the collection of universal support-forest-counting quantities obtained from the moments $M_r^{(2)}(T)$ for $1\le r\le \dim W_n$. The Laplacian spectrum is the spectrum in the $(n-1,1)$ representation, and the $(n-2,2)$-spectrum is the spectrum in the $(n-2,2)$ representation.

Spectral support-forest-profile conjecture. If $T_1$ and $T_2$ have the same Laplacian spectrum and the same $(n-2,2)$ support-forest profile, then $T_1$ and $T_2$ are isomorphic. Equivalently, no two non-isomorphic trees have identical $(n-1,1)$- and $(n-2,2)$-spectra.

For trees, the $(n-2,2)$-spectrum determines the support-forest profile, whose first nontrivial information beyond Laplacian data is a cubic combination of embedded forest counts. The conjecture asks whether the resulting hierarchy, together with Laplacian data, suffices for tree reconstruction.