Left orderability, foliations, and transverse $(pi_1,mathbb{R})$ structures for $3$-manifolds with sphere boundary

Let $M$ be a closed orientable irreducible $3$-manifold with a left orderable fundamental group, and $M_0 = M - Int(B^{3})$. We show that there exists a Reebless co-orientable foliation $mathcal{F}$ in $M_0$, whose leaves may be transverse to $partial M_0$ or tangent to $partial M_0$ at their intersections with $partial M_0$, such that $mathcal{F}$ has a transverse $(pi_1(M_0),mathbb{R})$ structure, and $mathcal{F}$ is analogue to taut foliations (in closed $3$-manifolds) in the following sense: there exists a compact $1$-manifold (i.e. a finite union of properly embedded arcs and/or simple closed curves) transverse to $mathcal{F}$ that intersects every leaf of $mathcal{F}$. We conjecture that $mathcal{F}$ is obtained from removing a $3$-ball foliated with horizontal disks from a taut foliation in $M$.