Cancellation conjecture for nonorientable 4-manifolds with cyclic fundamental group

Let $M_{2p}$ be a closed topological nonorientable $4$-manifold with cyclic fundamental group of order $2p$, where $p>1$ is odd. Theorem Cancellation Homeomorphism Classes asserts that $M_{2p}\mathbin{\#}2\cdot(S^2\times S^2)$ is homeomorphic to exactly one $4$-manifold in the specified collection, with the corresponding classification described by the theorem. Cancellation conjecture. The same conclusion should hold for $M_{2p}$ without any additional copies of $S^2\times S^2$. This would resolve the remaining cancellation problem needed for a complete classification of these manifolds; the source describes it as a difficult algebraic problem and does not state that it has been solved.