Martin and Thatte's maximum agreement subtree conjecture for balanced trees

Let ${\mathcal S}$ and ${\mathcal T}$ be balanced rooted binary phylogenetic $X$-trees, meaning that each has $2^m$ leaves for some non-negative integer $m$ and height $m$, and let $n=|X|$. Write ${\rm mast}({\mathcal S},{\mathcal T})$ for the maximum size of a common restriction that is an agreement subtree of ${\mathcal S}$ and ${\mathcal T}$. Martin and Thatte's conjecture.

$${\rm mast}({\mathcal S},{\mathcal T})\geq n^{\frac12}.$$

The conjecture proposed a square-root lower bound for the size of a maximum agreement subtree of two balanced rooted binary phylogenetic trees. It is refuted by the paper's main theorem, which gives, for every $c>0$, balanced rooted binary trees with maximum agreement subtree of size less than $c n^{\frac12}$; hence the conjecture is false.