All inequalities in this post are meant element-wise. Consider a stochastic $n \times n$ matrix $P$, i.e. $P \geq 0, P1 = 1$, where $1$ is a vector of ones. Consider a matrix $A$ of size $n \times m$, $A \geq 0$, whose span includes the row space of $P$. Therefore we have $AA^\dagger P^\top = P^\top$, where $\dagger$ denotes the Moore-Penrose pseudoinverse. Assume further that $A$ has independent columns and $m < n$.
Is it true that $A^\dagger P^\top A \geq 0$ ?
This question is a modification of this post, with an additional assumption on $A$.
Edit: I am also interested in the case where the structure of $A$ is additionally constrained as follows: The columns of $A$ contain a basis for the column space of $P^\top$, selected from the columns of $P^\top$, they can also contain other elementwise non-negative vectors.