  The decomposition of the four-point ABJM amplituhedron into negative
geometries produces a compact integrand of logarithmic of amplitudes such that
the infrared divergence only comes from the last loop integral, from which we
can compute the cusp anomalous dimension of the ABJM theory. In this note, we
integrate $L-1$ loop momenta of the $L$-loop negative geometries for all
four-loop negative geometries and a special class of all-loop ladder-type
negative geometries by a method based on Mellin transformation, and from these
finite quantities we extract the corresponding contribution to the cusp
anomalous dimension. We find that the infrared divergence of a box-type
negative geometry at $L=4$ is weaker than other negative geometries, then only
tree-type negative geometries contribute to the cusp anomalous dimension at
$L=4$. For the all-loop ladder-type negative geometries, we prove and
conjecture some recursive structures as integral equations in Mellin space and
find that they cannot contribute zeta values like $\zeta_3,\zeta_5$ to the cusp
anomalous dimension.
