Multivariate Matsumoto-Yor property is rather restrictive

Matsumoto and Yor [2001. An analogue of Pitman's 2M-X theorem for exponential Wiener functionals. Part II: the role of the GIG laws. Nagoya Math. J. 162, 65-86] discovered an interesting invariance property of a product of the generalized inverse Gaussian (GIG) and the gamma distributions. For univariate random variables or symmetric positive definite random matrices it is a characteristic property for this pair of distributions. It appears that for random vectors the Matsumoto-Yor property characterizes only very special families of multivariate GIG and gamma distributions: components of the respective random vectors are grouped into independent subvectors, each subvector having linearly dependent components. This complements the version of the multivariate Matsumoto-Yor property on trees and related characterization obtained in Massam and Wesołowski [2004. The Matsumoto-Yor property on trees. Bernoulli 10, 685-700].