  In this paper, we show that the Turaev-Viro invariant volume conjecture posed
by Chen and Yang is preserved under gluings of toroidal boundary components for
a family of $3$-manifolds. In particular, we show that the asymptotics of the
Turaev-Viro invariants are additive under certain gluings of elementary pieces
arising from a construction of hyperbolic cusped $3$-manifolds due to Agol. The
gluings of the elementary pieces are known to be additive with respect to the
simplicial volume. This allows us to construct families of manifolds with an
arbitrary number of hyperbolic pieces such that the resultant manifolds satisfy
an extended version of the Turaev-Viro invariant volume conjecture.
