The Colored Jones Function does not determine any colored Links--Gould polynomial

We prove that there is no uniform description of ${\mathrm{L}\mathrm{G}}^{(n)}$ in terms of colored Jones polynomials for all knots, even when allowing infinitely many colors and arbitrary operations. We prove the $n=1$ case, using an existing cabling formula and the separation of the Conway and Kinoshita–Terasaka knots by ${\mathrm{L}\mathrm{G}}^{(2)}$. We develop an odd cabling formula and use a Vandermonde argument to show that for each $n\ge 2$ some satellite of each of these knots is distinguished by ${\mathrm{L}\mathrm{G}}^{(n)}$.