that 11.3 is probably going to diminish... does it ever go higher?
I wouldn't rule out alternator ad a problem but being that the car is already NOT being charged, that is another problem that could affect your current one.
Same as before, see if you can pull up OBC 9.0 =
Get current reading with car running.
Then go to hood, find alternator, unplug sensor part, then go back to OBC see if it changes.
If your is already at 11.3 v while running, your alternator already isn't charging though so yeah.. that's your first problem!