The gotcha is right there in the Abstract: "solid carbon and gallium oxide are the final reaction products of this process."
This is the thermodynamic equivalent of rearranging the deckchairs on the Titanic. It achieves literally nothing except for shifting the oxygens around. They're still around, still bonded, but even harder. That's what this process doing -- reducing the carbon using a liquid metal. Reducing the gallium oxide back to the metal takes even more energy than was released by burning the C to CO2 in the first place! If you have an energy source that exceeds the world total carbon usage... then you don't need to burn carbon any more![1]
At best, this could be used to convert some magical future energy source like unlimited fusion power into pure carbon, eliminating the need for coal mining. But that's it.
This -- and any approach like this -- can never be practically used to counter the effects of CO2.
The finance equivalent would be: "We found this new way of producing $100 that only requires $110 in expenses!"
Actually not $110 at all. A sibling comment mentioned that "Gallium is cheap (~$300/kg)".
For comparison, coal is on the order of $150/ton. Not kilograms. Tons. Literally 2,000x cheaper than Gallium per unit weight!
So this paper proposes reducing $100 of carbon by using $200,000 of gallium.
Sure, you have gallium oxide left behind that can be recycled for a bit less than $300/kg, but certainly not for $150/ton!
[1] The few direct uses such as using coke to produce iron from its ore can be replaced by hydrogen gas. Similarly, groups are working on processes for iron similar to the one used for aluminium that requires only electricity as the input.