Let $M$ be a smooth compact connected surface, possibly with boundary. Let $f:M \rightarrow M$ be a diffeomorphism that fixes the boundary.
Assume that $f$ is pseudo-isotopic to the identity map, i.e. that there exists a diffeomorphism $F:M \times I \rightarrow M \times I$ that fixes $M \times \{0\}$ and $\partial M \times I$, and satisfy that $F|_{M \times \{1\} } = f \times 1$.
Is $f$ necessarily smoothly isotopic to the identity map? Is there a counter example?
I have only found a counter example in higher dimensions (see this answer, or the original paper), and I have not found any relevant references. It seems plausible to me that this will turn out to be true in low dimensions, but my intuition of pseudo-isotopies is somewhat limited.
To convert my comments to an answer: Pseudo-isotopy implies (boundary-preserving) homotopy. The latter (in dimension 2) implies smooth isotopy. See for instance
Earle, C. J.; Eells, J., A fibre bundle description of Teichmüller theory, J. Differ. Geom. 3, 19-43 (1969). ZBL0185.32901.
Earle, C. J.; Schatz, A., Teichmüller theory for surfaces with boundary, J. Differ. Geom. 4, 169-185 (1970). ZBL0194.52802.