Polynomial-time recognition conjecture for 0/1 slack matrices

Let $S\in\{0,1\}^{m\times n}$. A matrix is a slack matrix if it is the slack matrix of a polytope. Polynomial-time recognition conjecture. There is an algorithm polynomial in $m,n$ that correctly determines whether $S$ is the slack matrix of a polytope. The conjecture concerns the unresolved complexity of recognizing slack matrices among 0/1-valued matrices, equivalently the recognition problem for 2-level polytopes.