Yazdi's virtual Euler class one conjecture

Let $M$ be an oriented closed hyperbolic $3$--manifold. An even integral lattice point is a class $w\in H^2(M;{\mathbb R})$ whose integral lattice coordinates are even; suppose that $w$ has dual Thurston norm $1$. A finite cover is a covering $\tilde{M}\to M$ with finitely many sheets, and let $\tilde{w}$ denote the pullback of $w$ to $\tilde{M}$. A transversely oriented taut foliation is a taut foliation equipped with a transverse orientation.

Yazdi's virtual Euler class one conjecture. For any even integral lattice point $w\in H^2(M;{\mathbb R})$ of dual Thurston norm $1$, there exists some finite cover $\tilde{M}$ of $M$ such that $\tilde{w}$ is the real Euler class of some transversely oriented taut foliation on $\tilde{M}$.

This conjecture is a virtual version of Thurston's Euler class one conjecture, which the source says was disproved. The paper gives positive examples with first Betti number $2$ or $3$ and partial examples with first Betti number at least $4$; the general statement remains unresolved.