I did quite a few numerical  computations and think the following is true, but I cannot prove it.


Let $\varphi(x):=\sum_{i=1}^n \varphi_i(x_i)$ where $x=(x_1,...,x_N) \in \mathbb{R}^n$ and $\varphi_i$ be even scalar convex function such that $\varphi''$ is increasing on $[0,\infty).$

We then define a probability measure $$p_y(x) \propto e^{\langle y, x\rangle}e^{-\varphi(x) } \ dx. $$

Can we show that for all unit vectors $z \in \mathbb{R}^n$ we have for all $y \in \mathbb{R}^n$

$$Var_{p_0}(\langle z,x \rangle) \ge Var_{p_y}(\langle z,x \rangle)?$$

In other words, the variance of $\langle z,x\rangle$ where $x$ is distributed according to $p_y$ is maximized at $y=0.$

Is this a known theorem or somehow easy to show?