In the reference book on infinite divisibility and generalised gamma convolution, BONDESSON, Lennart. Generalized gamma convolutions and related classes of distributions and densities. Springer Science & Business Media, 2012., page 26, the theorem 2.4.3 states :
An MED is infinitely divisible and it's convolutional roots are MED's.
Unfortunately the proof is not that explicit. Is there a way to construct the mixture distribution of the convolutional roots ? I mean a explicit mapping wich map's the mother mixture distribution and the "root size" to the new mixture distribution ?