What is the maximum apogee a Crew Dragon can reach in a highly elliptical orbit (by both the Falcon 9 and the Falcon Heavy)? There is a planned touristic mission in which the current apogee record of 1,368 km (788 mi) set by Gemini 11 is to be broken.
Crew Dragon with Falcon Heavy can theoretically be sent to a lunar free return trajectory, so pretty much any apogee could be done. As for just a Falcon 9, it's hard to know for certain. Putting in a desired orbit of 2000 km with the NASA Performance Vehicle Calculator, I got the following plots. Note that 9000 km is the launch mass of Crew Dragon.
The problem is that the flight profile for a human rated spacecraft has the potential to interfere with this. I'm going to make the assumption that if you can do a RTLS, you can do a human rated trajectory with a drone ship landing. The real answer is probably somewhere between the lines.
So I'm going to guess they can set the record, but probably not by a huge margin. Definately less than 2000 km, probably around the 1400-1700 km range.