In the chosen parameterization, energy gap fluctuations are moderately non-Gaussian (with µ (3) = 0.33 and µ (4) = 0.22). The second order cumulant approximation fails to reproduce the correct vibronic progression, both underestimating the separation between vibronic peaks and predicting wrong intensities. When adding the third order correction to the lineshape, both peak intensities and positions are much improved, but at the cost of spurious unphysical spectral regions of negative absorbance. This spurious behavior of the truncated third order cumulant expansion is commonly observed for sufficiently non-Gaussian fluctuations. We note that, while deviating significantly from the exact lineshape, the second order cumulant approximation is guaranteed to yield a positive-definite and thus physically meaningful absorption spectrum for arbitrary non-Gaussian energy gap fluctuations (see SI Sec. II).