  In this paper, we study the variation of the Turaev--Viro invariants for
$3$-manifolds with toroidal boundary under the operation of attaching a
$(p,q)$-cable space. We apply our results to a conjecture of Chen and Yang
which relates the asymptotics of the Turaev--Viro invariants to the simplicial
volume of a compact oriented $3$-manifold. For $p$ and $q$ coprime, we show
that the Chen--Yang volume conjecture is stable under
$\left(p,q\right)$-cabling. We achieve our results by studying the linear
operator $RT_r$ associated to the torus knot cable spaces by the
Reshetikhin--Turaev $SO_3$-Topological Quantum Field Theory (TQFT), where the
TQFT is well-known to be closely related to the desired Turaev--Viro
invariants. In particular, our utilized method relies on the invertibility of
the linear operator for which we provide necessary and sufficient conditions.
