Check the Air Cutoff valve diaphragm.
Expensive little sucker.. Cover, spring and diaphragm/valve
Fill it with RTV??? Rubber tire tube to make a new diaphragm??
14 VALVE SET, AIR CUT 16048-MZ8-H00 In Stock $84.02 $65.02
Look up MR VTX Carb 101 for carb teardown pix.
I think a rubber tire tube will dissolve with gas, possibly contaminating the carb. Just a heads up. You'll want to use something fuel safe for this IMO.