  In the work by M. C. Lee, A. Naber, and R. Neumayer a beautiful
$\varepsilon$-regularity theorem is proved under small negative scalar
curvature and entropy bounds. In that paper, the $d_p$ distance for Riemannian
manifolds is introduced and the quantitative stability results are given in
terms of this notion of distance, with important examples showing why other
existing notions of convergence are not adequate in their setting. Due to the
presence of an entropy bound, the possibility of long, thin splines forming
along a sequence of Riemannian manifolds whose scalar curvature is becoming
almost positive is ruled out. It is important to rule out such examples since
the $d_p$ distance is not well behaved in the presences of splines that persist
in the limit. Since there are many geometric stability conjectures where we
want to allow for the presence of splines that persist in the limit, it is
crucial to be able to modify the $d_p$ distance to retain its positive
qualities and prevent it from being sensitive to splines. In this paper we
explore one such modification of the $d_p$ distance and give a theorem which
allows one to estimate the modified $d_p$ distance, which we expect to be
useful in practice.
