  The tableau reconstruction problem, posed by Monks (2009), asks the
following. Starting with a standard Young tableau $T$, a 1-minor of $T$ is a
tableau obtained by first deleting any cell of $T$, and then performing jeu de
taquin slides to fill the resulting gap. This can be iterated to arrive at the
set of $k$-minors of $T$. The problem is this: given $k$, what are the values
of $n$ such that every tableau of size $n$ can be reconstructed from its set of
$k$-minors? For $k=1$, the problem was recently solved by Cain and Lehtonen. In
this paper, we solve the problem for $k=2$, proving the sharp lower bound $n
\geq 8$. In the case of multisets of $k$-minors, we also give a lower bound for
arbitrary $k$, as a first step toward a sharp bound in the general multiset
case.
