They have the problem completely wrong
You can easily carry oxygen with you.
Liquid O2 1 liter bottle holds 1.1
kg of oxygen. According to the previous post you need 35
mg per breath. 1 liter bottle is going to last a while.
So the problem is you actually need more volume to breath, both as a carrier to fill your lungs with o2, and as a medium to get rid of co2 from your body.
So their purposed system would be way more useful if it could primarily extract a large volume of nitrogen from the water, rather than oxygen (or maybe some o2)......I think there is less nitrogen in water than oxygen though
What could also be useful is if they reversed the membrane function to remove C02, and reuse the carrier while adding some O2. This would be a new type of re-breather.
Or a combination of the two above or whatever.
Point is they probably "solved" the wrong problem even in fantasy land.