A criterion for virtual Euler class one

Let $M$ be an oriented closed hyperbolic $3$--manifold. Suppose that $w$ is a rational second cohomology class of $M$ with dual Thurston norm $1$. Upon the existence of certain nonvanishing Alexander polynomials, the author shows that the pullback of $w$ to some finite cover of $M$ is the real Euler class of some transversely oriented taut foliation on that cover. As application, the author constructs examples with first Betti number either $2$ or $3$, and partial examples with any first Betti number at least $4$, supporting Yazdi's virtual Euler class one conjecture.