Prove that a certain block combination of totally unimodular matrices is totally unimodular

Question

I conjecture the following.

Given three rectangular matrices $A$, $B$ and $C$ such that the following two block matrices $ \begin{bmatrix} A & B \\ \end{bmatrix} $ and $ \begin{bmatrix} A \\ C \\ \end{bmatrix} $ are totally unimodular, the following matrix $\begin{bmatrix} A & B \\ C & 0 \\ \end{bmatrix} $ is totally unimodular.

Can you help me to prove it?

Answer 1

Here is a counterexample, i'm sorry.

$ \det \left[ \begin{array}{c c | c} 1 & 0 & 1 \\ 0 & 1 & 1 \\ \hline -1 & -1 & 0 \end{array} \right] = 2 $