This hose is attached to carbs 3 & 4, meaning a float in carburetors 3 and/or 4 was stuck open. Often this means either dirt clogging the float needle seat, or the float needle and/or seat is too worn to seal properly. At 98k miles it's likely they're all worn beyond spec, so I decided to replace all 4 needles & seats.
The parts required are 4 x K&L 18-5082 economy carburetor repair kits, available online or at most Suzuki shops.
The needles & seats come in a little plastic capsule. Open the capsule and soak overnight in clean gas.
Remove the carburetors and clean them. Make sure everything is air-dried, then carefully remove the float by unscrewing the float screw. The following photo shows carburetor 3 with jets and floats removed for seat replacement.
With the float removed, unscrew the bolt holding down the seat. It may be quite difficult to get out. I used a manual impact screwdriver on a few of them.
Wiggle the seat out with a needle-nose pliers, then clean and air-dry the passage under the seat.
Press-fit the new seat, then very gently tap with a dead-blow hammer to make sure it's fully bottomed out.
Screw in the bolt, snug but not too tight.
Hang the new float needle onto the front of the float.
Note that the float needle hanger is directional. Orient the side with the two loops towards the float body (back of float).
Reinstall the float, then screw down the float bowl cover and move on to the next carburetor.
It's a bit hard to see due to the poor photo quality, but the white buildup around the old seat gasket is from ethanol in the gas--not good.
Reinstall the carburetors and check for smooth idle and leaks.
I'm happy to report the Bandit actually started and idled after this service!