Bootstrap tests and confidence regions for functions of the population covariance matrix have the desired asymptotic levels, provided model restrictions, such as multiple eigenvalues in the covariance matrix, are taken into account in designing the bootstrap algorithm.
"Bootstrap Tests and Confidence Regions for Functions of a Covariance Matrix." Ann. Statist. 13 (1) 95 - 115, March, 1985. https://doi.org/10.1214/aos/1176346579