In-folding convertible roof

A convertible roof (20) includes a segmented pair of front roof (46) rails that in-fold when being retracted, thereby reducing the packaging space required to stow the convertible roof (20). The in-folding of the segmented front roof rails (46) is controlled by linkages (48) between the segmented portions and the number one roof (44) bow and synchronizes and constrains the in-folding of the segmented front roof rails. A dampening device is utilized to dampen a portion of the movement of the top stack mechanism between extended and retracted positions.