  Fock-Goncharov's moduli spaces $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$ of framed
${\rm PGL}_3$-local systems on punctured surfaces $\frak{S}$ provide prominent
examples of cluster $\mathscr{X}$-varieties and higher Teichm\"uller spaces. In
a previous paper of the author (arXiv:2011.14765), the so-called ${\rm SL}_3$
quantum trace map is constructed for each triangulable punctured surface
$\frak{S}$ and its ideal triangulation $\Delta$, as a homomorphism from the
stated ${\rm SL}_3$-skein algebra of the surface to a quantum torus algebra
that deforms the ring of Laurent polynomials in the cube-roots of the cluster
coordinate variables for the cluster $\mathscr{X}$-chart for $\mathscr{X}_{{\rm
PGL}_3,\frak{S}}$ associated to $\Delta$. We develop quantum mutation maps
between special subalgebras of the cube-root quantum torus algebras for
different triangulations, and show that the ${\rm SL}_3$ quantum trace maps are
compatible under these quantum mutation maps. As a result, the quantum ${\rm
SL}_3$-${\rm PGL}_3$ duality map constructed in the previous paper is shown to
be independent of the choice of an ideal triangulation.
