Strong Shamai–Laroia conjecture for additive Gaussian interference

Let $h,p,v\in\mathbb{R}_+$, let $X_1^G\sim\mathcal{N}(0,p)$ and let $Z\sim\mathcal{N}(0,v)$ be independent of $X_1^G$. Let $X$ and $X_1$ be i.i.d. random variables with mean $0$ and variance $p$. Strong Shamai–Laroia conjecture. For all such $X$ and $X_1$,

$$I(X;X+hX_1^G+Z)\leq I(X;X+hX_1+Z).$$

The conjecture says that Gaussian interference minimizes the mutual information in this scalar ISI setting. Shlomo Shamai brought it to the authors' attention in a stronger form allowing arbitrary memory in the interference term; the general status is not specified here.