We can draw carbon from the atmosphere, sure. It's an inefficient process costing a lot of energy, but given they're making the final rings molecule by molecule, it isn't as if we'd need a lot.

...for the purposes of scrubbing CO2 from the air? There's absolutely no reason we'd use this as the final product. Graphite or diamond would do the trick just as well, and either is cheap to make.

I don't know the details of the "wet" chemistry step described, but I imagine it would be possible to extract CO2 from the atmosphere and transform it over many step to eventually get to the required precursors.

The practical reality is that each transformation step has imperfect yeilds and I would guess that the equipment required to do the molecule by molecule transformation in the last step would require significant energy to run.

So to attempt to this today, you would end up producing more CO2 into the atmosphere than you removed.