Zartash–Robeva conjecture on Gaussian convolution and log-supermodularity

Let $d$ be a positive integer, let $f\in L_1(\mathbb{R}^d)$ be log-supermodular, and define the standard Gaussian density

$$g(x)=(2\pi)^{-\frac{d}{2}}e^{-|x|^2/2}.$$

The convolution $f*g$ is defined by $(f*g)(x)=\int_{\mathbb{R}^d}f(x-y)g(y)\,dy$.

Zartash–Robeva conjecture. The convolution $f*g$ is log-supermodular.

The conjecture concerns the stability of log-supermodularity under convolution, a property that does not hold for arbitrary pairs of log-supermodular functions. The paper's abstract states that this conjecture is confirmed in the special case of the standard Gaussian density, so the conjecture is solved by the results of the paper.