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](https://arxiv.org/abs/hep-th/0202128), [Nadler](https://arxiv.org/abs/math/0612399) and [Tsygen](https://arxiv.org/pdf/0905.0290.pdf). I was wondering if there are more lurking about...