Smoothing topological pseudo-isotopies of 4-manifolds

Given a closed, smooth 4-manifold X and self-diffeomorphism f that is topologically pseudo-isotopic to the identity, we study the question of whether f is moreover smoothly pseudo-isotopic to the identity. If the fundamental group of X lies in a certain class, which includes trivial, free, and finite groups of odd order, we show the answer is always affirmative. On the other hand, we produce the first examples of manifolds X and diffeomorphisms f where the answer is negative. Our investigation is motivated by the question, which remains open, of whether there exists a self-diffeomorphism of a closed 4-manifold that is topologically isotopic to the identity, but not stably smoothly isotopic to the identity.