Part of the issue with the idea of "proving" the Church-Turing thesis is that the Church-Turing thesis isn't a precise mathematical statement. Rather, it's the idea, or "belief" if you will, that any model of computation that could feasibly be constructed is either equal in power to a Turing machine or weaker than a Turing machine.
If we were to try to formalize the Church-Turing thesis into a rigorous mathematical statement, we'd have to somehow give definitions to terms like "model of computation" or "feasibly constructed" that are, by design, very vague. After all, the whole point of the Church-Turing thesis is to claim that working with Turing machines really captures what computation is because any other model we could have picked would end up equal in power to Turing machines. We can precisely pin down particular rules for what a model of computation is and what it means for it to be feasible to construct, but by doing so we'd limit the resulting theorem to only talking about those classes of machines. If someone later on devises some new model of computation that is wildly different than everything we've ever thought about so far and ends up being more powerful than Turing machines, then the intuitive Church-Turing thesis would be false (Turing machines don't actually capture computation!) but the rigorous Church-Turing thesis would still be true (because this new model of computation wouldn't fit into the framework developed so far).
Here's another way to see this. There are formal proofs that Turing machines can simulate computers and that computers with unbounded memory can simulate TMs. We have similar results from certain types of cellular automata (Conway's Game of Life, for example), string rewriting systems (unrestricted grammars), crystal structures (see this amazing link), etc. All of these specific cases lend support to the Church-Turing thesis, but none of these statements, in isolation, is the Church-Turing thesis.
So suppose that we prove that a TM can simulate the human brain. This would be a huge result! But that by itself doesn't prove the Church-Turing thesis because, by nature, the Church-Turing thesis is unprovable. In a sense, the answer to the more general question of "can the Church-Turing thesis be proven?" is "no," so your question falls out as a special case.
That said, it is potentially possible to disprove the Church-Turing thesis by finding some model of computation that can solve a number of problems that TMs cannot but at the same time is still something that can be feasibly constructed in the universe. In that sense, the Church-Turing thesis is more of a falsifiable scientific hypothesis than a mathematical theorem.