The paper you linked to about the one learning algorithm hypothesis is from 2012. Since that time the theory has gained significant strength from the advances in DL, and in particular the work on deep reinforcement learning. Proving that an ANN with a relatively simple initial/prior architecture and about 1 million neurons can reach human-level performance on a set of 100 games when trained end to end with RL is pretty strong (albeit indirect) evidence for the one learning hypothesis.

One key remaining question is then: how does the brain actually implement approximate optimization/learning that is at least as good as back-prop? We know that back-prop is not biologically realistic. On that front, Bengio's group has made significant recent progress with a new technique/theory called target propagation 1, which originated in part as an explanation for how the brain could implement credit assignment, but it also shows promise as a potential replacement for backprop 2 - which further increases the biological plausibility.

In terms of more direct evidence, the hippocampus in particular appears to have a simple explanation in terms of reinforcement learning 3.

In terms of the prefrontal cortex in particular, there are working theories that explain much of the PFC as a set of modules specialized for working memory buffers that are controlled by gating units in the basal ganglia. That whole system in particular is also driven/learned through dopamine based RL.