This looks a lot like what I was thinking of. They don't provide much information about their extracting filter, but after the CO2 is extracted it is compressed into water creating carbonic acid, then injected it into a suitable basalt formation with an impermeable cap so that carbonate salts could form[1]. They claim all the injected CO2 could become carbonates in under 10 years[2].
If this works like they say, then it's cause for hope.