Saeki's rigidity conjecture for 4-dimensional manifolds admitting special generic maps

Let $M$ and $N$ be $4$-dimensional homeomorphic closed and connected manifolds, and suppose that each admits a special generic map into some Euclidean space of dimension smaller than $4$. Saeki's rigidity conjecture. Then $M$ and $N$ are diffeomorphic. This conjecture concerns the rigidity of the diffeomorphism type of $4$-dimensional manifolds admitting special generic maps into low-dimensional Euclidean spaces. It was attributed to an informal comment by Saeki in 2010; its resolution is not specified here.