In this paper, Jeremy England discusses about dissipation-driven adaptation, which proposes a mathematical explanation for the origin of life. While there is almost general consensus on the theoretical validity of his results, his interpretation of it remains unproven. I was wondering if any ideas for the experimental verification of this theory have been proposed so far.
Your phrase "Mathematical explanation for the origin of life" is not a good description of that paper. It's really a thermodynamic limit for the minimum amount of energy / entropy involved in self-replication.
The "explanation of the origin of life" (abiogenesis), when it is found, will involve mainly (1) chemistry, (2) the logic / math of Darwinian evolution. Yes, the explanation will also be enriched by insights from other fields, including thermodynamics and including this paper. But the heart of the explanation will be (1) and (2), and cannot come from pure physics.
Anyway, this paper is one of many "thermodynamic limit" results known to physicists, and they are all pretty much impossible to experimentally verify in a direct way. For example, Carnot's limit says that no engine can be more efficient than a certain formula. You could in principle disprove it by making an engine that surpasses the limit. But how could you prove it? After all, if you make an engine that's less efficient than the limit, it doesn't prove that the limit is correct, maybe you're just bad at making engines.
There are similar thermodynamic limits on the efficiency of solar cells, photochemistry, computer hardware, air conditioners, and on and on. They all similarly are not directly experimentally provable.