We study the maximally-helicity-violating (MHV) six-gluon scattering amplitude in planar $\mathcal{N} = 4$ super-Yang-Mills theory at finite coupling when all three cross ratios are small. It exhibits a double logarithmic scaling in the cross ratios, controlled by a handful of ``anomalous dimensions" that are functions of the coupling constant alone. Inspired by known seven-loop results at weak coupling and the integrability-based pentagon OPE, we present conjectures for the all-order resummation of these anomalous dimensions. At strong coupling, our predictions agree perfectly with the string theory analysis. Intriguingly, the simplest of these anomalous dimensions coincides with one describing the light-like limit of the octagon, namely the four-point function of large-charge BPS operators.