 Solution

Write ${\phi }_G={\phi }_{G,L}+{\phi }_{G,s}$. In the local-type primordial non-Gaussianity expansion, the term linear in the short field is ${\phi }_{G,s}+2f_{\mathrm{NL}}{\phi }_{G,L}{\phi }_{G,s}$. The long-field square contributes to the long background, the mean subtraction is spatially homogeneous, and the short-field square is excluded at the order requested. HenceThis is the long-short coupling measured by a squeezed bispectrum configuration.

Because the long potential is constant over the short-scale region, multiplication by $1+2f_{\mathrm{NL}}{\phi }_{G,L}$ commutes with the linear cosmological Poisson equation and any fixed linear smoothing filter. Thus the short density contrast satisfies ${\delta }_s=(1+2f_{\mathrm{NL}}{\phi }_{G,L}){\delta }_{G,s}$. Taking the conditional expected value over the short Gaussian random field givesThe short-scale variance modulation by local non-Gaussianity is conditional on the long mode; averaging over that mode as well would erase the term linear in its zero mean.