On the robustness of posterior means

Consider a normal location model $X\mid \theta \sim N(\theta ,{\sigma }^2)$ with known ${\sigma }^2$. Suppose $\theta \sim G_0$, where the prior $G_0$ has zero mean and variance bounded by $V$. Let $G_1$ be a possibly misspecified prior with zero mean and variance bounded by $V$. We show that the squared error Bayes risk of the posterior mean under $G_1$ is bounded, subjected to an additional tail condition on $G_1$, uniformly over $G_0,G_1,{\sigma }^2>0$.