Jenkinson–Morris conjecture on periodic Lyapunov-minimizing measures

Let $T:{\mathbb T}\to{\mathbb T}$ be a $C^k$ expanding self-map, where $k\geq 2$, and let $\mathcal{E}^k$ denote the space of such maps with the $C^k$ topology. A Lyapunov minimizing measure minimizes the Lyapunov exponent over invariant measures.

Jenkinson–Morris conjecture. For a generic $T\in\mathcal{E}^k$, the Lyapunov minimizing measure is unique and supported on a periodic orbit of $T$.

The paper proves this assertion in the $C^2$ topology, while the conjecture remains open for $k>2$.