  We show that a monotone Lagrangian $L$ in $\mathbb{CP}^n$ of minimal Maslov
number $n + 1$ is homeomorphic to a double quotient of a sphere, and thus
homotopy equivalent to $\mathbb{RP}^n$. To prove this we use Zapolsky's
canonical pearl complex for $L$ with coefficients in $\mathbb{Z}$, and various
twisted versions thereof, where the twisting is determined by connected covers
of $L$. The main tool is the action of the quantum cohomology of
$\mathbb{CP}^n$ on the resulting Floer homologies.
