Some casual searching suggests this is a problem with sulfur contamination of the tin-lead solder layer. I can’t vouch for any of that, as the flat areas forming the capacitor seem to be silver-plated, but …
After some flailing around, I completely disassembled the capacitor, applied 800 grit sandpaper to remove all of the solder / flux / corrosion / tarnish / surface plating from the pins, dabbed on some RMA flux, then applied a thin layer of solder to both sides. Fortunately, the capacitor could be disassembled; they don’t make ’em like that any more.
The solder layers must be thin, because the slots in the ceramic base must pass two or three pins apiece: four or six solder layers add too much thickness. Solder-wick is my friend!
For reference, the 700 pF side looks like this:
The steel washer does not have a mica washer underneath (as does the washer on the 400 pF right side). The two grayish steel plates go on the top.