We consider a family of Turaev-Viro type invariants for a $3$-manifold $M$ with non-empty boundary, indexed by an integer $r\geqslant3,$ and propose a volume conjecture for hyperbolic $M$ that these invariants grow exponentially at large $r$ with a growth rate the hyperbolic volume of $M.$ The crucial step is the evaluation at the root of unity $\exp({2\pi\sqrt{-1}}/{r})$ instead of that at the usually considered root $\exp({\pi\sqrt{-1}}/{r}).$ Evaluating at the same root $\exp({2\pi\sqrt{-1}}/{r}),$ we then conjecture that, the Turaev-Viro invariants and the Reshetikhin-Turaev-Lickorish invariants of a closed hyperbolic $3$-manifold $M$ grow exponentially with growth rates respectively the hyperbolic and the complex volume of $M.$ This uncovers a different asymptotic behavior of the values at other roots of unity than that at $\exp({\pi\sqrt{-1}}/{r})$ predicted by Witten’s Asymptotic Expansion Conjecture, which may indicate some different geometric interpretation of the Reshetikhin-Turaev invariants than the $SU(2)$ Chern-Simons theory. Numerical evidences are provided to support these conjectures.