A theorem of Chapman (Chapman, T. A. Homotopic homeomorphisms of infinite-dimensional manifolds. Compositio Math. 27 (1973), 135–140) states that every two open embeddings $f,g:M\to N$ of Hilbert manifolds (he states it more generally) which are homotopic are also isotopic in the sense that each level is an open embedding.
Is it also true in the smooth category? Is it true if we replace open embedding with submersion?