Conjectural equality of diagonal numerator-denominator gcds

Let $N(C_{n,2};s,t)$ and $D(C_{n,2};s,t)$, respectively $N(T_{n,2};s,t)$ and $D(T_{n,2};s,t)$, be the numerators and denominators of the corresponding bigraded Poincaré series.

Diagonal-gcd conjecture. For all $n$,

$$\gcd\left( N(C_{n,2};t,t),D(C_{n,2};t,t) \right)=\gcd\left( N(T_{n,2};t,t),D(T_{n,2};t,t) \right).$$

This is the fourth conjecture in the paper's list and is reported to hold for $n\le6$ by computation. Its validity for arbitrary $n$ is not resolved in the supplied source.