Conjecture on vanishing numerator values for trace algebras

Let $C_{n,2}$ and $T_{n,2}$ be the trace algebras with numerator polynomials $N(C_{n,2};s,t)$ and $N(T_{n,2};s,t)$ in their bigraded Poincaré series.

Vanishing-numerator conjecture. For $n\ge5$,

$$N(C_{n,2};1,1)=N(T_{n,2};1,1)=0.$$

The paper states that this conjecture is true for $n\le6$ and notes that it would imply that $C_{n,2}$ has no bigraded system of parameters for $n\ge5$. Its validity for general $n$ remains open in the supplied source.