Matrix Rank Polarization

Question

Assume the following promise problem. The input to this problem is a matrix $M \in \{0,1\}^{n \times n}$ that is promised to have rank either $n$ or $n-1$. The task is to come up with a transform $T:\{0,1\}^{n \times n} \rightarrow \{0,1\}^{n\times n}$ such that:

1. Each entry of the output of $T$ is a constant (independent of $n$) degree function of the input bits.
2. If $M$ has rank $n$, then $T(M)$ also has rank $n$.
3. If $M$ has rank $n-1$, then $T(M)$ has rank at most $n/2$.

Is there such a transform?