Boundedness of cumulant correction terms for the three-letter pair count

Let $T_r$ denote the correction to the $r$th cumulant of $X^{(n)}_3$ arising from dependence among the pair indicators. Cumulant-correction conjecture.

$$|T_.|=\mathcal{O}(1).$$

The surrounding argument explains that bounded corrections, in particular for the third and fourth cumulants, would suffice for the proposed Gaussian limiting distribution of $X^{(n)}_3$. The notation $T_.$ is schematic in the source, so the precise range of cumulant orders intended should be checked against the full paper.