Finite termination of the extended simplex procedure for non-completely-positive matrices

Let $A\in{\mathcal{S}}^n$ be a matrix that is not completely positive. A separating witness for $A$ is a matrix $W\in{\mathcal{COP}}^n$ such that $\langle A,W\rangle<0$. Procedure~ is the extended pivoting procedure described in the paper.

Finite-termination conjecture. For $A\notin{\mathcal{CP}}^n$, Procedure~ with a suitable pivot rule in Step~2(b) ends after finitely many iterations with a separating witness $W$.

The conjecture asserts finite termination with a certificate whenever the input lies outside the completely positive cone. The paper does not establish whether the procedure can fail to provide a separating witness after finitely many iterations, and leaves the existence of a suitable pivot rule open.