Maybe I'm just misunderstanding your use of the terms. I take it by "open loop" you mean that the AI would seek to generate an improved version of itself, but would simply provide that code back to the researcher rather than running it itself?

Roughly, yes. But I see recursive self-improvement as having a hardware component as well, so "closed loop" also includes giving the AI control over electronics factories and electronic assembly robots.

... it seems like it would be pretty hard to do research on self-improvement without a closed loop; isn't the expectation usually that the self-improvement process won't start doing anything particularly interesting until many iterations have passed?

Odd. My expectation for the software-only and architecture-change portion of the self-improvement is that the curve would be the exact opposite - some big gains early by picking off low-hanging fruit, but slower improvement thereafter. It is only in the exponential growth of incorporated hardware that you would get a curve like that which you seem to expect.