The 
[Littlewood conjecture](https://en.wikipedia.org/wiki/Littlewood_conjecture) is an example that meets all my requirements.

It is easy to state and obviously $\Pi^1_1$.  Furthermore [this comment](https://gowers.wordpress.com/2009/11/17/problems-related-to-littlewoods-conjecture-2/#comment-4395) gives me confidence that it has no known  reduction to an arithmetical sentence.