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 $M$