  We compute the asymptotical growth rate of a large family of $U_q(sl_2)$
$6j$-symbols and we interpret our results in geometric terms by relating them
to volumes of hyperbolic truncated tetrahedra. We address a question which is
strictly related with S.Gukov's generalized volume conjecture and deals with
the case of hyperbolic links in connected sums of $S^2\times S^1$. We answer
this question for the infinite family of fundamental shadow links.
