Let $X$ be a standard Normal variable. Let $\mathbb{E}$ denote expectation with respect to $X$.
Consider $$ R(\tau) := \inf_{T} \sup_{|\theta| \leq \tau} \mathbb{E}[(T(\theta + X) - \theta)^2] \quad \mbox{and} \quad R := \inf_{T} \sup_{\theta \in \mathbb{R}} \mathbb{E}[(T(\theta + X) - \theta)^2] $$ Above the infima range over measurable maps $T$.
Clearly, $\tau \mapsto R(\tau)$ is an nondecreasing map and $R(\tau) \leq R$. Is it true that $$ \lim_{\tau \to \infty} R(\tau) = R? $$