Let $H$ be a binary hypothesis class, it is easy to see that if $H$ is (efficiently) properly PAC learnable then it is also (efficiently) testable (here we use the standard notion of within or $\epsilon$-far testable).
My question is:
Fixed a distribution, is there any known hypothesis class that is hard (say NP-hard) to properly learn (to avoid the problem to be trivial, we assume the class is statistically PAC-learnable) under this specific distribution, but easy (say within polynomial time) to test under the same distribution?
As far as I know most of the results for proving hardness of learning are reduced to the hardness of testing (i.e. decide a gap version of the risk minimization is hard).