In this paper (arXiv:2305.01438) the quantum vacuum interaction energy between a pair of semitransparent two-dimensional plates represented by Dirac delta potentials and its first derivative, embedded in the topological background of a sine-Gordon kink is studied through an extension of the TGTG-formula (firstly discovered by O. Kenneth and I. Klich) to weak curved backgrounds. Quantum vacuum oscillations around the sine-Gordon kink solutions are interpreted as a quantum scalar field theory in the spacetime of a domain wall. Moreover, the relation between the phase shift and the density of states (the well-known Dashen-Hasslacher-Neveu formula) is also exploited to characterize the quantum vacuum energy.