Left orderability for surgeries on the $[1,1,2,2,2j]$ two-bridge knots

Let $M$ be a $\mathbb{Q}$-homology solid torus. In this paper, we give a cohomological criterion for the existence of an interval of left-orderable Dehn surgeries on $M$. We apply this criterion to prove that the two-bridge knot that corresponds to the continued fraction $[1,1,2,2,2j]$ for $j\geq 1$ admits an interval of left-orderable Dehn surgeries. This family of two-bridge knots gives some positive evidence for a question of Xinghua Gao.