Your probability measure is a product measure, so by 
$$\text{Var}_y(\langle z,X\rangle) = \sum_{i=1}^nz_i^2\text{Var}_{y_i}(X_i)$$
everything reduces to the 1d case. Let $q_y(dx)=e^{xy-\varphi(x)-C(y)}dx$ be one of the marginals, where $y\in\mathbb{R}$ and $C(y)$ is chosen such that $q_y$ is normalized, and denote by $U_y$ the 1d r.v. with distributon $q_y$. It can be shown that $C(y)-C(0)$ is the cumulant-generating function of $U_0$, and $\text{Var}_y(U_y)=C''(y)$. Thus the variance has a local maximum at $y=0$ iff the third cumulant $\kappa_0^{(3)}$ of $U_0$ vanishes and the fourth cumulant $\kappa_0^{(4)}$ is negative (which seems to be the case under your assumptions at least for $\kappa_0^{(3)}$ although I haven't checked for $\kappa_0^{(4)}$ but this should follow from the sub-Gaussianity of $q_0$). At least this shows that your conjecture is plausible. I don't know how to show the global maximum.*

(*) It seems that $C$ also gives cumulants of $q_y$. Unfortunately I don't have time now but I'll maybe come back later and try to complete this in case there are no other attempts.