This is a slightly vague question (for which I apologize in advance): can somebody give examples to open conjectures on the behavior of the $Fuk(M,\omega)$ that come from string theory and can be phrased in a mathematically (semi)-rigorous way?

Two examples I know of are Kontsevich's co-sheaf conjecture, and the connection between Deformation quantization and the Fukaya category in Bressler and Soibelman, Nadler and Tsygen. I was wondering if there are more lurking about...