Olivier, you took out the wrong connector!
It's the yellow wires that get hot and melt the brown connector, which is what I've taken out on both Futura and Tuono (thanks FB!)..
Do the yellow wires feel hot after 10 minutes? If not, it's okay.
Doing my nephew's Falco this weekend. Measured it against my Tuono last week:
Battery at rest: 12,83V (both)
Idle with lights on: Falco 13,75V - Tuono 14,30V
4k rpm lights on: Falco 13,12V - Tuono 14,12V!
Measuring is to be done at the battery, not at the dash.