A recent paper by Villegas (1969) uses a fiducial type argument to lend support to the use of Jeffrey's invariant prior for the variance-covariance matrix in the non-informative situation. Sampling is assumed to be on a multi-normal random variable. Villegas proceeds by selecting a pivotal quantity which has a fixed distribution. Since there is nothing unique about this pivotal quantity, we note in this paper, that the Villegas fiducial approach could lead to other priors, unless more restrictions are imposed. In Section 2, we construct an example involving a "lower triangular" decomposition of a Wishart matrix variable that has the distribution of the disguised Wishart variable of Tan and Guttman (1971). Interestingly, an "upper triangular" decomposition of the same Wishart matrix leads to yet another prior.
"Inconsistencies in the Villegas Method of Determining a Prior Distribution." Ann. Statist. 2 (2) 383 - 386, March, 1974. https://doi.org/10.1214/aos/1176342674