It’s not an unsolved physics problem. Every satellite in space has to deal with it and even the ISS deals with it by having massive radiator arrays that face perpendicular to the sun.

The problem with data centers in space is one of materials science and engineering: how to make radiators large enough and effective enough to cool it while also being economically feasible, both in terms of construction and getting them up there in the first place.

We can make a space data center right now. It would just be terrible and expensive.

The physics problem regarding radiator arrays isn't unsolved, but it's not a problem that scales up gracefully. Small-scale radiators could get by via passive cooling, but large-scale radiators need active cooling, and now you need fluid, pipes, and pumps that all represent additional launch mass and points of failure (and the pumps are generating heat of their own, so now you need more radiators...).

I think it's not necessarily about being the cheapest option, but a more politically acceptable one. I don't think you're going to get people protesting a data center in space considering it won't be next to their house, won't use water, and won't lead to increased electricity rates. I could see companies paying a premium to keep the political heat associated with traditional data centers off their backs.