  An update is given on our long-term effort to perform a three-loop check on
the Affleck-Alvarez-Manton/Lebedev-Ritus exponentiation conjecture for the
imaginary part of the Euler-Heisenberg Lagrangian, using 1+1 dimensional QED as
a toy model. After reviewing the history and significance of the conjecture, we
present trigonometric integral representations for the single electron loop
contributions to the three-loop Lagrangian, and develop a symmetry-based method
for the calculation of their weak-field expansion coefficients.
