The effect of intramolecular excluded volume interactions on the conformations of isolated comb-shaped copolymers-"molecular bottle brushes＇＇-in athermal solvents confined between two hard impenetrable parallel plates is studied by off-lattice Monte Carlo simulations. Equilibrium conformational properties of molecules consisting of a freely jointed backbone of 100 hard spheres (beads), where 50 equally flexible side chains of lengths M up to 30 beads are connected, are determined as a function of the width of the slit, including the pure two-dimensional (2D) case. Besides general properties such as the size and the shape of the chains and the orientation of the chains with respect to the confining plates, the persistence length of the bottle-brush backbone lambda is considered in particular. It is demonstrated that due to the side chains, the backbone is, in fact, effectively confined between two soft plates, i.e., the side chains act like softening springs between the backbone and plates. The most striking result is that, unlike the three-dimensional (3D) situation, in the 2D case the ratio between lambda and the diameter D, lambda/D, of the bottle brush starts to increase as a function of the side chain length for M greater than or equal to 10. For 3D, an increase of lambda/D (the essential parameter for lyotropic behavior) as a function of M is predicted by theory but has so far not been observed numerically for the regime studied (i.e., M less than or equal to 30). These results suggest that due to the weaker excluded volume effect in 3D, a possible upturn of lambda/D may only be observed for longer side chains than studied so far.