Tight small Seifert fibered manifolds with $e_0=-2$

In this paper we provide the classification of tight contact structures on some small Seifert fibered manifolds. As an application of this classification, combined with work of Lekili in \cite{L2010}, we obtain infinitely many counterexamples to a question of Honda-Kazez-Mati'{c} that asks whether a right-veering, non-destabilizable open book necessarily supports a tight contact structure.