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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 $G_0$ haszero mean and unit variance. Let $G_1$ be a possibly misspecified prior withzero mean and unit variance. We show that the squared error Bayes risk of theposterior mean under $G_1$ is bounded, uniformly over $G_0, G_1, \sigma^2 > 0$.

Mean-variance constrained priors have finite maximum Bayes risk in the normal location model