No, it will not. The output transformer core is magnetized mostly by the current surge that charges the parafeed capacitor, which occurs before the power tube begins to conduct. The only solution is to short the output transformer primary winding with a relay, which opens up after everything is stabilized.
In the best of all worlds, I would add another PC board containing a regulated 6.3vDC supply for the driver heater, and three time delay relays to sequentially turn on the high voltage supply, then the soft start, and finally the output transformer. Among other problems, I have yet not found a place to put such a board without sticking out below the wood base or interfering with other components!