Equality of relative-entropy, Riemannian and geodesic contraction coefficients

Let ${\cal G}$ be the specified class of metrics, let $g\in{\cal G}$, and let $\phi$ be a stochastic map for which the contraction coefficients below are defined. The quantities $\eta_g^{\rm RelEnt}(\phi)$, $\eta_g^{\rm Riem}(\phi)$, $\eta_g^{\rm geod}(\phi)$, and $\eta_g^{\rm Dobrushin}(\phi)$ denote the relative-entropy, Riemannian, geodesic, and Dobrushin contraction coefficients, respectively.

Equality conjecture. For each fixed $g\in{\cal G}$,

$$\eta_g^{\rm RelEnt}(\phi)=\eta_g^{\rm Riem}(\phi)=\eta_g^{\rm geod}(\phi)\leq\eta_g^{\rm Dobrushin}(\phi).$$

The conjecture asserts that the three contraction coefficients agree and are bounded above by the Dobrushin coefficient; the paper presents it among conjectures that had already been discussed, without supplying a resolution.