Mean-square consistency for Gaussian stationary processes

Let $X$ be a Gaussian stationary process with covariance function $RX$, and let $S_J[\mathcal P_J]X$ denote its windowed scattering transform over the path set $\mathcal P_J$. It is a mean-square consistent estimator of $\overline S X$ when

$$\lim_{J\rightarrow\infty}E\left(\left\|S_J[\mathcal P_J]X-\overline S_JX\right\|^2\right)=0.$$

Gaussian consistency conjecture. If $X$ is a Gaussian stationary process with $\|RX\|_1<\infty$, then $S_J[\mathcal P_J]X$ is a mean-square consistent estimator of $\overline S X$. Mean-square consistency implies convergence in probability and almost-sure convergence as stated in the source, while the supplied passage reports numerical observations for a large class of ergodic processes rather than a proof of this Gaussian-process claim.