Suppose $\mu,\nu$ are probability measures on $\mathbb{R}^d$ such that for all $z \in \mathbb{R}^d$, for all $g:\mathbb{R} \to \mathbb{R}$ convex,
$$ \int g(x \cdot z) \mu(dx) \leq \int g(x \cdot z) \nu(dx), $$
where $x\cdot y =\sum_{i=1}^d x_i y_i$ is the Euclidean inner product. Does it follow that $\mu$ and $\nu$ are in convex order, i.e. for all $f:\mathbb{R}^d \to \mathbb{R}$ convex,
$$ \int f(x) \mu(dx) \leq \int f(x) \nu(dx)? $$