We prove that for a suitable (open) class of open, smoothly bounded domains in the cotangent bundle of a surface of genus $g \geq 2$ any exact symplectomorphism is homotopic to one which is smooth up to the boundary. In particular, such boundaries are not unseen in the sense of Eliashberg–Hofer. The contact boundaries of these domains have Anosov Reeb flows. The methods employed include the properties of such flows in three dimensions, and analysis of two-dimensional transverse flow maps along the lines of Thurston, and thus appear restricted to the case of three-dimensional contact boundaries.