There is something else to try before baking your phone!
I just revived my FP3 from sudden death (most likely) with the folded paper method. I was about to try the oven reflow when I remembered a trick from the forum that had already helped me with a different issue on the FP2: putting a few layers of paper in between to re-establish contact of the pins.
Most likely its just a temporary fix, but my phone has been back to life for a few hours now - enough to retrieve all data. I will edit this post with updates on the longevity of the solution.
What I did was putting 4 layers of paper between the metal shield of memory/cpu and the plastic frame. On the other side, I put 3 layers between the metal shield of the power management chips and the metal thingy that is screwed over the mainboard. There is definitely room for experimentation here, thats just what worked first try for me.
Thank you for being this extremely helpful community and good luck to all the ones with broken phones!