  The well-posedness of a generalized Coleman--Gurtin equation equipped with
dynamic boundary conditions with memory was recently established by C.G. Gal
and the author. Additionally, it was established by the author that the problem
admits a finite dimensional global attractor and a robust family of exponential
attractors in the case where singularly perturbed memory kernels defined on the
interior of the domain and on the boundary of the domain coincide. In the
present article we report advances concerning the asymptotic behavior of this
heat transfer model when the memory kernels do {\em not} coincide. In this
setting we obtain a weak exponential attractor whose basin of attraction is the
entire phase space, that is, a finite dimensional exponentially attracting
compact set in the weak topology of the phase space. This result completes an
analysis of the finite dimensional attractors for the generalized
Coleman--Gurtin equation equipped with dynamic boundary conditions with memory.
