Minimality conjecture for non-reconstructible pedigrees

Let $\mathcal{T}(X_0)$ be a pedigree of order $n$. A pedigree is $(n-1)$-reconstructible if it is determined up to congruence by the collection of its restrictions to subsets of $n-1$ individuals in $X_0$. Minimality conjecture. If $\mathcal{T}(X_0)$ is not $(n-1)$-reconstructible, then it has depth at least $n-2$, and it has at least $2^{n-1}$ ancestors at depth $n-2$. The conjecture asserts that the counterexample constructed in the cited theorem is minimal; its resolution is not specified in the source.