= Solution
Complete the square in the independent normal prior and <likelihood>:
$$
\tau\theta_i^2+\tau_\varepsilon(X_i-\theta_i)^2
=(\tau+\tau_\varepsilon)\left(\theta_i-\frac{\tau_\varepsilon X_i}{\tau+\tau_\varepsilon}\right)^2
+\frac{\tau\tau_\varepsilon}{\tau+\tau_\varepsilon}X_i^2.
$$
Thus, conditional on the full observations, the hospital parameters remain independent and
$$
\boxed{\theta_i\mid X\sim
N\!\left(\frac{\tau_\varepsilon}{\tau+\tau_\varepsilon}X_i,\
\frac1{\tau+\tau_\varepsilon}\right)}.
$$
If $j$ is the observed minimum index and $X_j=x$, its conditional distribution is exactly this formula with $X_j=x$. No extra truncation is introduced because the selection index is already a function of the conditioned data. Its posterior mean is shrunk toward zero, so the worst measured performance is not simply identified with the worst true performance.
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