I am told by a very well-placed source that the text of the Motu Proprio is being printed at the Vatican Tipografia.
On Italian news via RAI tg2 on Sunday, it was said that the MP will come "in the upcoming days" and "next week".
It has been said that the MP would come before the Holy Father leaves Rome for his summer break. He leaves in early July.
UPDATE: The Roman daily Il Tempo has a story on the MP. It states that the Pope will release the MP "in settimana… during the week". Once again, we find the comment that at least 30 people have to make the request.