Every time I would take the part of the pen that the converter plugs into out of the ultrasonic cleaner, a little bit of (still quite dark) ink would would drip out.

I finally figured out what I was doing wrong. It turns out that this part is so narrow that the surface tension of the water was causing an air bubble to get trapped inside, preventing the feed from being properly cleaned. I solved it by manually injecting some soapy water with a syringe inside to blow the air bubble out and it cleaned right out afterward.

Still, that's two hours of my life I'm not gonna get back.