Failure of the fully marked surface theorem for C2 taut foliations

Let $M$ be a closed hyperbolic 3-manifold, let $\mathcal{F}$ be a $C^2$ taut foliation on $M$, and let $S$ be a fully marked surface. $C^2$ failure conjecture. There exists such a triple $(M,\mathcal{F},S)$ for which no $C^2$ taut foliation $\mathcal{G}$ on $M$ has oriented plane field homotopic to that of $\mathcal{F}$ and has $S$ homologous to a union of leaves. The conjecture reflects the expected failure of the theorem's analogue in the $C^2$ category, where the flexibility of Denjoy blow-up is unavailable.