Guys,
do not swap those tubes until you have solved your PSU problems! As far as I know many of you are using the original rondo or edcor transformers, and they are both way underspecified. I mean seriously, why didn't I open my mouth when you guys were arranging the edcor group buy, at least the disaster could have been partially averted. If I remember correctly those transformers were specified at exactly the total heater requirement for two channels. This is very bad! You need like twice or perhaps 1.5X of more power for safety headroom to prevent issues exactly like many of you have seen.
Those transformers will barely power up the heaters of two channels. The issue is indeed the 5687 tube and the very high 900mA requirement. If several of them decide to eat a whole 1A of current (which is entirely possible) the transformer might already be over saturated, near breaking point. Danger zone.
Get another heater transformer and you will be able to run all four of the 5687 tubes. Don't swap in any alternatives like ECC99 or 6N6P. Their heaters are something like 700-800mA and it will not help much, if at all.