Non left-orderable surgeries on L-space twisted torus knots

We show that if $K$ is an L-space twisted torus knot $T_{p,pk\pm 1}^{l,m}$ with $p\ge 2$, $k\ge 1$, $m\ge 1$ and $1\le l\le p-1$, then the fundamental group of the $3$-manifold obtained by $\frac{r}{s}$-surgery along $K$ is not left-orderable whenever $\frac{r}{s}\ge 2g(K)-1$, where $g(K)$ is the genus of $K$.