If it takes 6.8V to get the one weak tube to "perk up"- then it DEFINITELY has filament sensitivity, and the cathode is definitely worn out and on the way out. Especially given how much it drops in output, when all four tubes are in (where, if my calculations of the internal heater impedance being just over 0.16 ohms or so are correct- the heater drop will be about 0.6V which would make it wind up right at exactly 6.3V with all four tubes in)...
I bet that if you put it in a normal tube tester, and turn the heater voltage down to 5.0V- the emission of the tube will drop, to pretty close to zero...
Regards,
Gordon.