Same thing with my 15 year old LT2000. I did the fuel filter, was fine. Thought the carburetor and installed a new one, just the other day. Still same thing for 1/2 tank when used to run till empty.
Not sure what the hose line from the front to the air box does. Assume just a breather/vent.
Is a mystery for myself and Google searched a bit as well. Was hoping the solenoid or such. I haven't check the red wire to the carburetor yet. Sounds like a fuel shut-off, I may disconnect and see would still run or not.