Plurisubharmonicity conjecture for the Teichmüller metric

Let $\mathcal{T}_g$ be Teichmüller space and let $T\mathcal{T}_g$ be its tangent bundle, equipped with the complex structure induced from the holomorphic tangent bundle. The Teichmüller metric is the Finsler norm on $T\mathcal{T}_g$. Teichmüller metric plurisubharmonicity conjecture. The Teichmüller metric is strictly plurisubharmonic on $T\mathcal{T}_g$. Strict plurisubharmonicity would give a strong complex-analytic convexity property for the metric on the tangent bundle; the source does not state a resolution.