It will be shown that if a confidence procedure is admissible when the criterion is probability of not covering the true value and expected length (or something more general than length), then the same confidence procedure is at least almost admissible when the criterion is probability of not covering the true value and probability of covering false values. The result is true (under mild conditions) for virtually all confidence region estimation problems.
"Admissibility Implications for Different Criteria in Confidence Estimation." Ann. Statist. 1 (2) 363 - 366, March, 1973. https://doi.org/10.1214/aos/1176342375