The taut-foliation extension conjecture for punctured 3-manifolds

Let $M$ be a closed orientable irreducible $3$-manifold such that $\pi_1(M)$ is left orderable, and let $M_0=M-\operatorname{Int}(B^3)$. Let $\mathcal{F}$ be a foliation of $M_0$ constructed in Theorem 1. Taut-foliation extension conjecture. There exists a taut foliation $\mathcal{F}_1$ in $M$ such that $(M_0,\mathcal{F})$ is obtained from removing a $3$-ball from $(M,\mathcal{F}_1)$; this also implies that $\mathcal{F}_1$ has a transverse $(\pi_1(M),\mathbb{R})$ structure. The conjecture proposes that the foliations produced on punctured manifolds extend over the removed ball, providing a weak form of the L-space conjecture and the related question about taut foliations.