Yes, but on the contrary, it could also be that Nechung is jealous and hence creating all sorts of problems for Dorje Shugden now. Just look at what Nechung has advised for stopping the practice of Dorje Shugden (
http://dorjeshugden.com/wp/?p=3883)
The Dalai Lama’s “investigations” involve invoking Nechung through the human State Oracle, who has had the following things to say (perhaps some of these were given in a false trance? It is difficult to know.):
1st answer of the State Oracle: “Dorje Shugden a powerful deity, only to be worshipped by beings with high realizations. However worshipping this deity would upset Goddess Palden Lhamo (a superior protecting deity, who does not have an oracle)”
2nd answer of the State Oracle: “the deity is appropriate to be worshipped by an individual, but not by a group”
3rd answer of the State Oracle: “Dorje Shugden is a deity, suitable to the others, but not to the successor of the 5th Dalai Lama and those working for the Gaden Phodrang Government established by the 5th Dalai Lama.”
4th answer of the State Oracle: “Dorje Shugden is a spirit born out of a Kagyupa-monk who hated the Tibetan government, and not the incarnation of Tulku Dragpa Gyaltsen”
5th answer of the State Oracle: “Dorje Shugden is the spirit of Tulku Dragpa Gyaltsen, whose Samaya bond to the 5th Dalai Lama was not good, thus it is harmful for this government.”
6th answer of the State Oracle: “Tulku Dragpa Gyaltsen was a good lama, whose works of composition are praiseworthy, therefore Dorje Shugden cannot be the spirit of such a master.”
7th answer of the State Oracle: “Tulku Dragpa Gyaltsen himself was a false Tulku, who came to be among the candidates for the 5th Dalai Lama and failed to be chosen, but through clever tactics of his mother on the first Panchen Lama Chokyi Gyaltsen, he was recognized as the fourth reincarnation of Panchen Sonam Dragpa (the teacher of 3rd Dalai Lama), but was then born as an evil, trouble-making spirit to harm the Tibetan government.”