The square-root agreement-subtree conjecture for balanced phylogenetic trees

Let $T_1$ and $T_2$ be balanced, rooted, binary trees of height $m$. An agreement subtree is a common subtree of the two trees on a shared set of leaves. The square-root agreement-subtree conjecture. Any two balanced, rooted, binary trees of height $m$ have an agreement subtree on at least

$$2^{m/2}$$

leaves. This would establish the conjectured square-root exponent for the maximum agreement subtree problem in the rooted balanced case; the statement is presented as a belief in the source and no resolution is given there.