Conjecture on binomial denominators for trace-algebra Poincaré series

Let $C_{n,2}$ and $T_{n,2}$ be the trace algebras whose bigraded Poincaré series have numerators $N(C_{n,2};s,t)$ and $N(T_{n,2};s,t)$ and denominators $D(C_{n,2};s,t)$ and $D(T_{n,2};s,t)$.

Binomial-denominator conjecture. The denominators $D(C_{n,2};s,t)$ and $D(T_{n,2};s,t)$ can be written as products of binomials $1-s^a t^b$, where $a$ and $b$ are nonnegative integers.

This is the first of four conjectures proposed from computations of the relevant rational Poincaré series. The paper reports that all four conjectures hold for $n\le6$, but does not establish the assertion in general.