On the Euler class one conjecture for fillable contact structures

In this paper, it is proved that every oriented closed hyperbolic $3$--manifold $N$ admits some finite cover $M$ with the following property. There exists some even lattice point $w$ on the boundary of the dual Thurston norm unit ball of $M$, such that $w$ is not the real Euler class of any weakly symplectically fillable contact structure on $M$. In particular, $w$ is not the real Euler class of any transversely oriented, taut foliation on $M$. This supplies new counter-examples to Thurston's Euler class one conjecture.