my aug-31 to sep-17 Star-rtw did include:
* Thursday evening: ORD-COS (overnight stay, more than 4 hours, less than 24 hours, not taking the next connecting flight)
* Friday morning: COS-DEN (next connecting flight, less than 4 hours) - ASE (more than 4 hours, not next connecting flight)
* Friday afternoon/evening: ASE - DEN (next connecting flight, less than 4 hours) - COS (one night stay, more than 4 but again less then 24 hours, not taking the next connecting flight)
* Saturday morning: COS - DEN (next connecting flight, less than 4 hours) - ASE (next connecting flight, less than 4 hours) - DEN (less than 4 hours, not next connecting flight) - SFO

I discussed the stopover-rules/itinerary several time with my (swiss based) LH-angel and finally we compromised that she would tolerate my two COS-stops (both being less than 24 hours, but more than 4 hours and never taking the next connecting flight). The compromise was, that the first COS stop would count as a stopover, but not the second one ...

We were both not 100% sure, if on a international ticket, with some entirely inner-North American segments 'woven' into it, the 24 hours or the 4 hours stopover rules should be applied. So we compromised and it worked out.

LH didn't ask UA for their opinion about this, they just booked it and that was it.