  We study the compact monotone Fukaya category of $T^*S^n$, for $n\geq 2$, and
show that it is split-generated by two classes of objects: the zero-section
$S^n$ (equipped with suitable bounding cochains) and a 1-parameter family of
monotone Lagrangian tori $(S^1\times S^{n-1})_\tau$, with monotonicity
constants $\tau>0$ (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 $S^n$ or one of the $(S^1\times S^{n-1})_\tau$. In the case of $T^*S^3$,
the monotone Lagrangians $(S^1\times S^2)_\tau$ can be replaced by a family of
monotone tori $T^3_\tau$.
