  We demonstrate how the matrix-valued mode-coupling theory of the glass
transition and glassy dynamics in planar confinement converges to the
corresponding theory for two-dimensional (2D) planar and the three-dimensional
bulk liquid, provided the wall potential satisfies certain conditions. Since
the mode-coupling theory relies on the static properties as input, the
emergence of a homogeneous limit for the matrix-valued intermediate scattering
functions is directly connected to the convergence of the corresponding static
quantities to their conventional counterparts. We show that the 2D limit is
more subtle than the bulk limit, in particular, the in-planar dynamics
decouples from the motion perpendicular to the walls. We investigate the
frozen-in parts of the intermediate scattering function in the glass state and
find that the limits time $t\to \infty$ and effective wall separation $L\to 0$
do not commute due to the mutual coupling of the residual transversal and
lateral force kernels.
