On the robustness of posterior means

Consider a normal location model $X \mid θ\sim N(θ, σ^2)$ with known $σ^2$. Suppose $θ\sim G_0$, 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 $G_0, G_1, σ^2 > 0$.