In the paper "Spin glasses and Stein's method", Sourav Chatterjee established Stein's equation for mixtures of two Gaussian densities in $\mathbb{R}$, which takes the form as in page 15 of https://statweb.stanford.edu/~souravc/talk_spin.pdf.

Is this form extendable to mixtures of two Gaussian densities in $\mathbb{R}^d$?

In the paper "Spin glasses and Stein's method" (https://arxiv.org/pdf/0706.3500.pdf), Sourav Chatterjee established Stein's equation for mixtures of two Gaussian densities in $\mathbb{R}$, which takes the form as in page 15 of https://statweb.stanford.edu/~souravc/talk_spin.pdf.

Is this form extendable to mixtures of two Gaussian densities in $\mathbb{R}^d$?