Carbon capture is the dumbest and most expensive way to move forward. It would be cheaper to just turn every coal plant into a solar farm then it would be to capture the carbon out of the air so why even bother. Oh yeah, corporate interests.
Trees are cheap but need a lot of space and water. People are cutting down forests because they want to use the space for something else.
Ocean surface algae also collect solar power and carbon, and there is little demand for ocean surface for other uses. Algae are limited by available trace minerals -- cheap ones.
The only one ongoing I know of is Sahara dust being (naturally) blown out to the Atlantic ocean. Otherwise, just talk. Somebody who did an experiment had his results suppressed, crudely, with a false narrative promoted in its place. Why that was seen as necessary, and how it happened, seem even more interesting than the experiment.