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The Littlewood conjecture is an example that meets all my requirements.

It is easy to state and obviously $\Pi^1_1$. Furthermore this comment gives me confidence that it has no known reduction to an arithmetical sentence. (Hopefully it even lacks a known reduction to a $\Sigma^1_1$ sentence.)

The Littlewood conjecture is an example that meets all my requirements.

It is easy to state and obviously $\Pi^1_1$. Furthermore this comment by Christian Reiher gives me confidence that it has no known reduction to an arithmetical sentence. (Hopefully it even lacks a known reduction to a $\Sigma^1_1$ sentence.)

The Littlewood conjecture is an example that meets all my requirements.

It is easy to state and obviously $\Pi^1_1$. Furthermore this comment gives me confidence that it has no known reduction to an arithmetical sentence. Hopefully it even lacks a known reduction to a $\Sigma^1_1$ sentence.

The Littlewood conjecture is an example that meets all my requirements.

It is easy to state and obviously $\Pi^1_1$. Furthermore this comment gives me confidence that it has no known reduction to an arithmetical sentence. (Hopefully it even lacks a known reduction to a $\Sigma^1_1$ sentence.)

The Littlewood conjecture is an example that meets all my requirements.

It is easy to state and obviously $\Pi^1_1$. Furthermore this comment gives me confidence that it has no known reduction to an arithmetical sentence.

The Littlewood conjecture is an example that meets all my requirements.

It is easy to state and obviously $\Pi^1_1$. Furthermore this comment gives me confidence that it has no known reduction to an arithmetical sentence. Hopefully it even lacks a known reduction to a $\Sigma^1_1$ sentence.