Well, thats more info. I'm thinking....look into that foam thing. Air filter would be my first guess. If it gobbled a chunk, best you know what you are looking for. As far as the battery, do you have access to a LOW amp charger? Like 2 amps or less? Bike batteries don't do well with high output chargers. Kind of a kiss of death. If it needs water, add DISTILLED water, not tap water. French kiss of death.
Did he indicate if the foam getting sucked in and the engine noise he heard when he shut it down were related? Anyways, I see a carb clean in your future. However, before I treated the carburetor, I would be checking for a sucking chest wound in the engine!
The Columbo in me takes interest in the fact he put in a new "timing chain". I see the left engine cover coming off to check things out. Maybe pulling the valve cover. Neither are difficult. Got a digital camera handy?