Nonexistence of a mapping class group-invariant earthquake metric on Teichmüller space

Let $\mathcal{T}$ be Teichmüller space, and let a mapping class group-invariant asymmetric metric on $\mathcal{T}$ mean an asymmetric metric preserved by the mapping class group action. A left earthquake is the earthquake deformation obtained by applying the left earthquake map to points of $\mathcal{T}$.

Nonexistence conjecture. There is no asymmetric metric on Teichmüller space which is mapping class group invariant and for which left earthquakes are geodesics.

The claim asks whether Teichmüller space can admit an asymmetric, mapping class group-invariant metric whose geodesics are precisely left earthquake paths. The surrounding discussion relates this possibility to horocyclic geometry and to Busemann's characterization of metrics with horospheres as shortest lines; the source provides no resolution of the claim.