Let $(H_k)_{k\in\mathbb{N}}$ be the sequence of hermite polynimials, $Z\sim N(0,1)$ and $G\in L^2(\mathbb{R},\phi)$ with $\operatorname{E}\left[Z\right]=0$. By $\phi$ we denote the density of the standard normal distribution. Than exists a unique representation \begin{align} G(Z)=\sum_{k=1}^{\infty}\frac{J(k)}{k!}H_k(Z) \end{align} in $L^2(\Omega)$ where \begin{align} J(k):=\operatorname{E}\left[G(Z)H_k(Z)\right]. \end{align} We call \begin{align} m:=\min_{k\in\mathbb{N}_{\geq1}}\operatorname{E}\left[G(Z)H_k(Z)\right]\neq 0\} \end{align} the hermite rank of $G$.
Assume now that the hermite rank of $G$ is 1. Is the hermite rank of \begin{align*} 1\{G(x)\leq u\} \text{ for fixed $u\in\mathbb{R}$} \end{align*} also 1? I.e. is $\operatorname{E}\left[1\{G(Z)\leq u\}H_1(Z)\right]= \operatorname{E}\left[1\{G(Z)\leq u\}Z\right]\neq 0$?