Left-orderability in Dehn fillings of pseudo-Anosov mapping tori

For pseudo-Anosov mapping tori with co-orientable invariant foliations and monodromies reversing their co-orientations, a family of taut foliations was constructed in previous work on Dehn fillings with all rational slopes outside a neighborhood of the degeneracy slope. In this paper, we prove that all such Dehn fillings have left-orderable fundamental groups. We present two approaches, both based on an analysis of the branching behavior from such taut foliations. The first approach produces an $\mathbb{R}$-covered foliation arising from this family for each filling slope, and the second approach shows that, depending on the choice of a suitable system of arcs on $\mathrm{\Sigma }$, one obtains a foliation that either has one-sided branching or is $\mathbb{R}$-covered. Consequently, the second approach associates to each Dehn filling a family of representations of its fundamental group into ${\mathcal{G}}_{\mathrm{\infty }}$, the group of germs at infinity, whereas the first approach yields an explicit left-invariant order. As an application, combining our results with earlier work in the literature, we verify the L-space conjecture for all surgeries on the $(-2,3,2k+1)$-pretzel knot ($k⩾3$) in $S^3$.