  This work is the second part of a program initiated in arXiv:2111.13258
aiming at the development of an intrinsic geometric well-posedness theory for
Hamilton-Jacobi equations related to controlled gradient flow problems in
metric spaces. Our main contribution is that of showing that the comparison
principle proven therein implies a comparison principle for viscosity solutions
relative to smoother Hamiltonians, acting on test functions that are mere
cylindrical functions of the underling squared metric distance and whose
rigorous definition is achieved from the Evolutional Variational Inequality
formulation of gradient flows (EVI). In particular, the new Hamiltonians no
longer require to work with test functions containing Tataru's distance. This
substantial simplification paves the way for the development of a comprehensive
existence theory.
