Practically, this means the methane fed into the chemical vapor deposition process has to be extracted from biogas (say, a chicken manure digester) or directly synthesized from air and hydrogen (Sabatier process). This feedstock will contain the same trace amount of carbon-14 as the atmosphere and will be incorporated into the resulting diamond.
This seems like such an obvious thing to do for various reasons. It allows the (increasingly irrelevant) natural diamond industry to easily distinguish between their product and synthetic products. Not only that, there's probably a huge market for such 'atmospheric diamonds' among the people interested in green products, or tech products. Finally, while the cost of synthesizing methane feedstock is still quite a bit higher than just using natural gas, that's negligible relative to the cost of the final product and shouldn't be a factor.
Time to go write up a business plan, isn't it?
P.S. full paper here: