Are there hypothesis classes that are hard to learn but easy to test? 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).
Take the class  $\mathcal{M}$ of monotone boolean functions under the uniform distribution on $\{0,1\}^n$: 


it is known that $O(\sqrt{n}/\varepsilon^2)$ queries are sufficient to test it (even with non-adaptive testers) [KhotMinzerSafra15].
learning $\mathcal{M}$ under the uniform distribution, even allowing membership queries, requires $2^{\Omega(\sqrt{n}/\varepsilon)}$ queries ([BshoutyTamon96,BlaisCOST15]).