  It is well-known that on-shell maximally helicity-violating gluon scattering
amplitudes in maximally supersymmetric Yang-Mills theory are dual to a bosonic
Wilson loop on a null-polygonal contour. The light-like nature of the intervals
is a reflection of the mass-shell condition for massless gluons involved in
scattering. Presently, we introduce a Wilson loop prototype on a piecewise
curvilinear contour that can be interpreted in the T-dual language to
correspond to nonvanishing gluon off-shellness. We analyze it first for four
sites at one loop and demonstrate that it coincides with the four-gluon
amplitude on the Coulomb branch. Encouraged by this fact, we move on to the
two-loop order. To simplify our considerations, we only focus on the Sudakov
asymptotics of the Wilson loop, when the off-shellness goes to zero. The latter
serves as a regulator of short-distance divergences around the perimeter of the
loop, i.e., divergences when gluons are integrated over a small vicinity of the
Wilson loop cusps. It does not however regulate conventional ultraviolet
divergences of interior closed loops. This unavoidably introduces a
renormalization scale dependence and thus scheme dependence into the problem.
With a choice of the scale setting and a finite renormalization, we observe
exponentiation of the double logarithmic scaling of the Wilson loop with the
accompanying exponent being given by the so-called hexagon anomalous dimension,
which recently made its debut in the origin limit of six-leg gluon amplitudes.
This is contrary to the expectation for the octagon anomalous dimension to
rather emerge from our analysis, suggesting that the current object encodes
physics different from the Coulomb branch scattering amplitudes.
