  We prove that there exists an open and dense subset $\mathcal{U}$ in the
space of $C^{2}$ expanding self-maps of the circle $\mathbb{T}$ such that the
Lyapunov minimizing measures of any $T\in{\mathcal U}$ are uniquely supported
on a periodic orbit.This answers a conjecture of Jenkinson-Morris in the $C^2$
topology.
