We show that a monotone Lagrangian $L$ in $\mathrm{CP}n$ of minimal Maslov number $n+1$ is homeomorphic to a double quotient of a sphere, and thus homotopy equivalent to $\mathrm{RP}n$. To prove this we use Zapolsky's canonical pearl complex for $L$ with coefficients in $Z$, 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 $\mathrm{CP}n$ on the resulting Floer homologies.