Nýdl's reconstructibility conjecture for trees

For an $n$-vertex graph, the $m$-deck is the multiset of its unlabeled induced $m$-vertex subgraphs, and a family of graphs is weakly $celltext{ell}$-reconstructible if no two graphs in the family have the same $(n-celltext{ell})$-deck. Nýdl's conjecture. For $ncellge 2celltext{ell}+1$, no two $n$-vertex trees have the same $(n-celltext{ell})$-deck; equivalently, trees with at least $2celltext{ell}+1$ vertices are weakly $celltext{ell}$-reconstructible. The conjecture is refuted by two $13$-vertex trees with the same $7$-deck, although the stated bound is sharp in general because examples with $2celltext{ell}$ vertices and the same $celltext{ell}$-deck are known.