How short can stationary charged scalar hair be?

It is by now well established that charged rotating Kerr-Newman black holes can support bound-state charged-matter configurations which are made of minimally coupled massive-scalar fields. We here prove that the externally supported stationary charged scalar configurations cannot be arbitrarily compact. In particular, for linearized-charged-massive-scalar fields supported by charged rotating near-extremal Kerr-Newman black holes, we derive the remarkably compact lower bound (r_field-r₊)/(r₊-r_-)>1 /s² on the effective lengths of the external charged scalar "clouds" [here r_field is the radial peak location of the stationary scalar configuration, and {s ≡J /M²,r_±} are, respectively, the dimensionless angular momentum and the horizon radii of the central supporting Kerr-Newman black hole]. Remarkably, this lower bound is universal in the sense that it is independent of the physical parameters (proper mass, electric charge, and angular momentum) of the supported charged scalar fields.