= Solution
The <Rao-Blackwell theorem> suggests conditioning $S$ on $T$. Part (d) gives
$$
\widetilde S
=\mathbb E[X_1^2-1\mid T]
=\frac12+\frac{T^2}{4}-1
=\frac{T^2}{4}-\frac12.
$$
It remains unbiased. Since $T\sim N(2\mu,2)$ and a normal variable with mean $m$ and variance $v$ satisfies $\operatorname{var}(T^2)=2v^2+4m^2v$,
$$
\operatorname{MSE}(\widetilde S)
=\frac1{16}\operatorname{var}(T^2)
=\frac1{16}(8+32\mu^2)
=\boxed{\frac12+2\mu^2}.
$$
This is strictly below $2+4\mu^2$ for every $\mu$. The calculation is recorded as the <Rao-Blackwell estimator of a squared normal mean>.
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