Wendl's uniqueness conjecture for fillings of unit cotangent bundles of surfaces

Let $Y$ be the unit cotangent bundle of a closed orientable surface $\Sigma_g$ of genus $g$, equipped with its standard contact structure $\xi_{std}$. A Stein or exact filling of $(Y,\xi_{std})$ is a compact symplectic filling of this contact manifold, and the disk cotangent bundle is the standard filling.

Wendl's conjecture. The diffeomorphism type of a Stein or exact filling of $(Y,\xi_{std})$ is unique and is given by the disk cotangent bundle for every $g$.

For $g=0,1$, uniqueness up to symplectic deformation equivalence is known, while for $g>1$ no exact or Stein filling not diffeomorphic to the disk cotangent bundle had been found. The statement concerns the stronger classification of fillings up to diffeomorphism.