  The asymptotic behavior of quantum $6j$-symbols is closely related to the
volume of truncated hyperideal tetrahedra\,\cite{C}, and plays a central role
in understanding the asymptotics of the Turaev-Viro invariants of
$3$-manifolds. In this paper, we propose a conjecture relating the asymptotics
of the discrete Fourier transforms of quantum $6j$-symbols on one hand, and the
volume of deeply truncated tetrahedra of various types on the other. As
supporting evidence, we prove the conjecture in the case that the dihedral
angles are sufficiently small, and provide numerical calculations in the case
that the dihedral angles are relatively big. A key observation is a
relationship between quantum $6j$-symbols and the co-volume function of deeply
truncated tetrahedra, which is of interest in its own right. More ambitiously,
we extend the conjecture to the discrete Fourier transforms of the Yokota
invariants of planar graphs and volume of deeply truncated polyhedra, and
provide supporting evidence.
