Consistency conjecture for the generalized Hermite-process spectral density estimator

Let $Z(x,t)$ satisfy Condition B and the previous assumptions, and let $F_q({\Greekmath 0115}_m)$ and $\widehat{F}_q({\Greekmath 0115}_m)$ be the corresponding population and sample quantities, with $q>(1-2\widetilde{d}_{\ast})^{-1}$. The frequencies are ${\Greekmath 0115}_m$ as defined above. Consistency conjecture. Under Condition B and the previous assumptions and notations on $Z(x,t)$, the estimator is consistent in probability:

$$\frac{\widehat{F}_{q}({\Greekmath 0115}_m)}{F_{q}({\Greekmath 0115}_m)}\mathrel{\longrightarrow_p}1.$$

This is proposed as an analogue of Theorem 1 of Robinson (1994) for the generalized, nonlinear Hermite-process setting. The conjecture concerns consistency of the spectral-density estimator at the low frequency ${\Greekmath 0115}_m$; the paper does not provide a resolution.