  We compute the null cusp anomalous dimension of ABJM theory at strong
coupling up to two-loop order. This is done by evaluating corrections to the
corresponding superstring partition function, weighted by the $AdS_4\times
\mathbb{CP}^3$ action in AdS light-cone gauge. We compare our result, where we
use an anomalous shift in the $AdS_4$ radius, with the cusp anomaly of
$\mathcal{N}=4$ SYM, and extract the two-loop contribution to the non-trivial
integrable coupling $h(\lambda)$ of ABJM theory. It coincides with the strong
coupling expansion of the exact expression for $ h(\lambda)$ recently
conjectured by Gromov and Sizov. Our work provides thus a non-trivial
perturbative check for the latter, as well as evidence for two-loop
UV-finiteness and quantum integrability of the Type IIA $AdS_4\times
\mathbb{CP}^3$ superstring in this gauge.
