It should hold to well past 15 PSI
Take the carb back off and remove the metering diaphram and the cover so you can see the needle and lever.
with fuel or WD 40 in the chamber, pump up the fuel inlet and look for bubbles around the needle you need to do this several times and manually kick the lever to release pressure to see if the needle seats and seals every time.
Watch the fuel and or WD 40 spray when you kick the lever or if it pops off on it's own too soon as it sprays quite far and can blind you pretty easily if you are not careful.
If the needle walls are worn the needle can tilt and not seal well in the seat.
This kind of wear was common on solidly mounted carbs on engines where the engine vibrations are transmitted directly into the carb. (the 306 carb is mounted right to the metal reed plate)
If you can't get the carb to hold any pressure and the needle is sealing properly the problem is in the governor.
The carb should hold 15 lbs or better and the needle should not drop at all.
The governor is the brass screw right behind the choke shaft but they are difficult to remove as the threads are glued to seal the threads.....don't bother to try to remove it unless you have too.