Both one-sided and two-sided interval estimators are examined using conditional criteria and we find that most common intervals have acceptable conditional properties. In the two-sided case we further examine three well known intervals and find the shortest-unbiased (Neyman-shortest) interval possessing the strongest conditional properties, with the minimum-length interval a close second. In the one-sided case we have the somewhat surprising result that the lower confidence interval (which results from inverting the UMP test of $H_0: \sigma \leq \sigma_0$) has weaker conditional properties than the upper interval (where a UMP test does not exist).
"Conditional Properties of Interval Estimators of the Normal Variance." Ann. Statist. 15 (4) 1372 - 1388, December, 1987. https://doi.org/10.1214/aos/1176350599