Conjectural relation between denominators of two trace algebras

Let $D(C_{n,2};s,t)$ and $D(T_{n,2};s,t)$ denote the denominators of the bigraded Poincaré series of the trace algebras $C_{n,2}$ and $T_{n,2}$.

Denominator-relation conjecture. For all $n$,

$$(1-s)(1-t)D(C_{n,2};s,t)=(1-s^n)(1-t^n)D(T_{n,2};s,t).$$

The paper reports that this relation holds for $n\le6$, based on the computations presented there. No proof or general resolution is supplied.