We've done a lot of playing around with various means to reduce microphony. I think it's a three part program.
First, keep the energy from getting into the gear. Keep gear away from speakers and out of room modes and put it on a stable shelf if possible. Use some kind of squishy stuff as feet under the gear. Sorbothane works, and Herbie's Tenderfeet are very nice. I've used folded bubble wrap in a pinch.
Second, try to keep energy from moving from the chassis into the tube. A silicone o-ring under the tube can help as long as the tube pins still make good contact with the socket. Damping the chassis panel itself can be helpful.
Third, and maybe the most important, damp the tube envelope itself. IME mass is your friend. A lot of you have seen the vinyl dipped lead rings I use on my BeePre. Something similar should work well on the 4pills. Maybe some lead tape or long thin fishing weights wrapped in Teflon tape and bent into a circle fit on the tube like a crown? Stay with temperature resistant materials. One user here tried the vinyl covered rings on a Paramount, which runs the tubes at much higher temps than BeePre, and got a nasty surprise when the vinyl started to cook. I have found that Herbie's dampers work pretty well at higher frequencies, and that a really massive weight seems to be most effective for lower stuff. Filament/heater length will probaby have some bearing on where the resonances occur and thus what kind of treatment will work best. Small tube, higher frequency resonance. Big tube, lower.