I suppose if it was stuck hard enough, it could. But I'm no expert.
I found a similar issue on my P229, but I didn't experience any failures. I had read a story about a police officer/detective that had gotten killed when his holstered SIG slid off a stack of papers he was carrying, hit the floor and discharged. The firing pin safety should have prevented it from happening, but apparently it was determined that he was using a holster not designed for his pistol. In order to get the retention snap to close, he would pull the trigger and thumb the hammer forward. He would keep pressure on the hammer as he holstered and snapped the strap. This resulted in the hammer not returning to its neutral position and kept the firing pin safety disengaged.
Anyway, I as took my pistol out to recreate those circumstances (unloaded, of course) I found my firing pin was sticking out of the hole. A little nudge and it would return. I don't know how long it had been broken, and it didn't take much force to recess it. Just pressing against a cartridge would suffice.
So I guess it depends on which requires more force: pushing the broken pin back in, or igniting the primer.
Of course, the XCR has a bolt where the firing pin hole is recessed, instead of a pistol with a flat breech face. So I think I would expect stuck pin would induce a failure to feed before an AD. In other words, I'd expect the pin to catch the rim of a cartridge preventing the case from sitting where the pin could line up with the primer. Still, if the stars aligned, I could see a kaboom.
Again, that's just my completely inexpert analysis.