  We prove a self-improvement property of a capacity density condition for a
nonlocal Hajlasz gradient in complete geodesic spaces. The proof relates the
capacity density condition with boundary Poincar\'e inequalities, adapts
Keith-Zhong techniques for establishing local Hardy inequalities and applies
Koskela-Zhong arguments for proving self-improvement properties of local Hardy
inequalities. This leads to a characterization of the Hajlasz capacity density
condition in terms of a strict upper bound on the upper Assouad codimension of
the underlying set, which shows the self-improvement property of the Hajlasz
capacity density condition.
