We prove new comparison results between the Wasserstein distance and its sliced and max-sliced counterparts. First, we show that the Hölder exponent d+22 obtained by Bobkov and Götze for the max-sliced 1-Wasserstein distance on the unit ball is optimal for every d≥2, settling a question raised in their work. Second, we show that sharper comparisons are possible under stronger structural assumptions: if ν
Nearby in the stack
is a discrete measure and the optimal coupling between
μ
and
ν
transports each point to a nearest atom of
ν
, then
Wp(μ,ν)≤CdKSWp,1(μ,ν)
for a universal constant
C
, where the complexity parameter
K
is always at most the number of atoms
N
and can be substantially smaller. This complements a similar bound due to Park and Slepčev. An analogous bound holds for the sliced Wasserstein distance based on
k
-dimensional projections. Finally, using a construction from geometric discrepancy theory due to Chen and Travaglini, we prove that the linear dependence on
K
in this bound cannot be improved, up to polylogarithmic factors.