Recently, results regarding the Inverse Design problem for Conservation Laws and Hamilton-Jacobi equations with space-dependent convex fluxes were obtaine. More precisely, characterizations of attainable sets and the set of initial data evolving at a prescribed time into a prescribed profile were obtained. Here, we present an explicit example that underlines deep diff erences between the space-dependent and space-independent cases. Moreover, we add a detailed analysis of the time asymptotic solution of this example, again underlining diff erences with the space-independent case.