Thurston's Euler class realization conjecture for taut foliations

Let $M$ be a closed orientable irreducible atoroidal 3-manifold with positive first Betti number. An integral class is a class $a\in H^2(M;\mathbb{R})$ arising from integral cohomology, and the parity condition means that $a$ lies in the image of $2H^2(M;\mathbb{Z})\to H^2(M;\mathbb{R})$. A taut foliation $\mathcal{F}$ has Euler class $e(\mathcal{F})\in H^2(M;\mathbb{R})$. Thurston's Euler class realization conjecture. For any integral class $a\in H^2(M;\mathbb{R})$ of dual Thurston norm one and satisfying the parity condition, there exists a taut foliation $\mathcal{F}$ of $M$ with Euler class $e(\mathcal{F})$ equal to $a$. The conjecture asks whether every eligible boundary class of the dual Thurston norm ball is realized by a taut foliation; the parity condition is necessary for Euler classes of transversely oriented plane fields.