Let $\mathfrak{D}$ denote the set of $n\times n$, trace-one, positive semi-definite matrices (known as density matrices in quantum information theory). Consider a Completely Positive Trace Preserving (CPTP) map $\mathcal{T}\colon\mathfrak{D}\to \mathfrak{D}, \ \rho\mapsto \mathcal{T}(\rho)$, of the form $$ \mathcal{T}(\rho)=\Pi \rho \Pi + \mathcal{T}^\perp (\rho)\qquad (*) $$ where $\Pi$ is a $n\times n$ orthogonal projection. Note that $\mathcal{T}^\perp (\Pi \rho\Pi)=0$ since $\mathcal{T}$ is trace preserving.
My question is the following one:
Given the CPTP map in $(*)$, is it true that $\mathrm{rank}(\Pi\rho\Pi)\leq \mathrm{rank}(\mathcal{T}(\rho))$, for all $\rho\in\mathfrak{D}$?
My guess is yes, and I think that the proof should exploit the fact that the map is (completely) positive. But I don't know how to proceed. Any help will be very appreciated!