How robust are your tight-binding model parameters?

In a previous post I discussed the problem of extracting reliable parameters for tight-binding (and Hubbard) models from ab initio band structure calculations. My comments then were influenced by the figure below, which has now appeared on the arXiv in a short review by Anthony Jacko.

First, the band structure for a specific organic material was calculated using a density functional theory (DFT) based approximation. The energy dispersion relations were then fit to a tight-binding model involving 8 different hopping integrals, t0, t1, ....t7. 

The horizontal axis indexes the 8 integrals, the vertical axis shows their values determined from a range of different fits, using slightly different fitting methods and different runs of the fitting algorithm. 

Note the significant differences.
Thus, caution is in order if one uses the common practise of simply performing one fit [which may look impressive to the naked eye].
Jacko notes that this is like getting the elephants trunk to wiggle.

As Jacko stresses in the review, the most reliable and physically transparent way to determine the tight-binding hopping integrals is to construct Wannier orbitals and then directly calculate the integrals. The results for that procedure are shown in green. The red curve is the global best fit.