Third-moment Kolmogorov bound conjecture for constrained U-statistics

Consider the setting of Theorem Trate: a constrained or unconstrained U-statistic built from the random sequence and kernel in the paper, with Kolmogorov distance $d_K$ and the theorem's variance and nondegeneracy assumptions. Third-moment Kolmogorov bound conjecture. Theorem Trate holds assuming only the third-moment condition $\textup{(A}_{3}\textup{)}$ instead of assuming that $f$ is bounded; consequently, the theorem's $O(n^{-1/2})$ Kolmogorov-distance estimate remains valid under $\textup{(A}_{3}\textup{)}$. The source notes that the analogous result is known for unconstrained symmetric U-statistics, while the extension to the more general setting considered here is conjectural.