  Given a quasisymmetric homeomorphism $\varphi$ of the circle, Bonsante and
Schlenker proved the existence and uniqueness of the minimal Lagrangian
extension $f_\varphi:\mathbb{H}^2\to\mathbb{H}^2$ to the hyperbolic plane. By
previous work of the author, its maximal dilatation satisfies $\log
K(f_\varphi)\leq C||\varphi||$, where $||\varphi||$ denotes the cross-ratio
norm. We give constraints on the value of an optimal such constant $C$, and
discuss possible lower inequalities, by studying two one-parameter families of
minimal Lagrangian extensions in terms of maximal dilatation and cross-ratio
norm.
