  It is well known that on arbitrary metric measure spaces, the notion of
minimal $p$-weak upper gradient may depend on $p$.
  In this paper we investigate how a first-order condition of the
metric-measure structure, that we call Bounded Interpolation Property,
guarantees that in fact such dependence is not present.
  We also show that the Bounded Interpolation Property is stable for pointed
measure Gromov Hausdorff convergence and holds on a large class of spaces
satisfying curvature dimension conditions.
