If B6 never fell back to 0V, then we can conclude that the 6080 is working properly, the socket is working properly, the solder joint at B5 is good, the solder joint at B4 is good (despite your 0V reading at B4, which isn't possible based on all other information), and that the path from ground, through the 3K resistor, and to the terminal strip is working on the headphone jack side of the amp, but not the volume pot side of the amp. This also confirms that the 12AU7 is doing what it needs to do to bias both halves of the 6080 properly.
This is absolutely a flaky connection, likely at terminal 3 or terminal 9.
It might help to triple check the resistance between terminal 3 and terminal 9, to make sure that terminals 9 and 10 don't have any debris between them that's conducting, etc.