If the Gilera factory had survived I think we would have seen a 500cc GFR.
If you look at the frame you will see a few things that seem to have no reason.
Hold a Saturno engine up against it and they begin to make sense.
Difficult to get official confirmation of that but after the factory had closed some of the Gilera guys went to Piaggio. I met one really great guy named Cantini (not sure about the spelling now) several times, he did not speak English but we chatted (difficulty) about a few things and he did suggest that the intention had been to make a new 500 / 560 using the GFR frame.
Sad end to this story, one day he was leaving the factory on an RC600 I think and was in collision with a car and he died. A very friendly and knowledgeable guy.
I came, I saw, I did a bit of biking.