Monotone Lagrangians in cotangent bundles of spheres

We study the compact monotone Fukaya category of T⁎Sn, for n≥2, and show that it is split-generated by two classes of objects: the zero-section Sn (equipped with suitable bounding cochains) and a 1-parameter family of monotone Lagrangian tori (S1×Sn−1)τ, with monotonicity constants τ>0 (equipped with rank 1 unitary local systems). As a consequence, any closed orientable spin monotone Lagrangian (possibly equipped with auxiliary data) with non-trivial Floer cohomology is non-displaceable from either Sn or one of the (S1×Sn−1)τ. In the case of T⁎S3, the monotone Lagrangians (S1×S2)τ can be replaced by a family of monotone tori Tτ3.