  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.
