Is a co-rotating Dark Disk a threat to Dark Matter directional detection?

Recent N-body simulations are in favor of the presence of a co-rotating Dark Disk that might contribute significantly (10%-50%) to the local Dark Matter density. Such substructure could have dramatic effect on directional detection. Indeed, in the case of a null lag velocity, one expects an isotropic WIMP velocity distribution arising from the Dark Disk contribution, which might weaken the strong angular signature expected in directional detection. For a wide range of Dark Disk parameters, we evaluate in this Letter the effect of such dark component on the discovery potential of upcoming directional detectors. As a conclusion of our study, using only the angular distribution of nuclear recoils, we show that Dark Disk models as suggested by recent N-body simulations will not affect significantly the Dark Matter reach of directional detection, even in extreme configurations.