I had a similar problem with a Philharmonc the high  shortwave band cutout at the low end.  The cause is either is "Q" dropped to low in the local oscillator coil or the transconductance of the local oscillator is to low.  I suspected originally since the Philharmonic uses multiple 6j5's scott selected the ones with highest emission.  Now 80 years later I think the shellac may have collected moisture lowering the coils "Q".  Now you could buy a bunch of 6j5's and select the  best one.  The problem is the price of 6j5's is through the roof!  I decided to think out of the box and use a later design tube. So what I did was mount a a 9 pin tube socket to a 8 pin octal plug.  I wired the 9 pin tube socket for a 12au7/12at7 types of tube.  I plugged the adapter into 6j5 local oscillator tube and gave it try with several 12au7's and 12at7's.  They  ALL worked fine so I used the 12au7 since is a low "emu" tube like a 6j5 only about 30% higher.  Now to apply the same thing to a "56" tube you will need an external 6 volt to power the 12au7/12at7.  You could try a 6 volt  battery to see if it works before adding a 6 volt filament transformer to the power supply or just buy a large bunch of 56's.