Extension of the Hermite-process limit theorem to higher-order integrals

Let $p\geq 3$. Under the setting and assumptions of Theorem nCLT, let $(X_k)$ be the stationary sequence and $(a_n)$ the normalization sequence defined there. The processes are considered in $D[0,1]$ with the uniform metric.

Higher-order limit conjecture. The conclusion of Theorem nCLT continues to hold if $p\geq 3$; that is, as $n\to\infty$,

$$\left(\frac{1}{a_n}\sum_{k=1}^{\floor{nt}}X_k\right)_{t\in[0,1]}\Rightarrow\left(\mu^p(f)Z_{p,\beta}(t)\right)_{t\in[0,1]}.$$

The cases $p\geq3$ are not established by the paper's method of moments because moment determinacy for the limiting multiple Gaussian integral either fails or is unknown; the conjecture extends the stated limit theorem beyond the cases $p=1$ and $p=2$.