The paper actually discusses a different convergence, namely that of the basis set size. While the abstract does state that "(the) transmission functions converge toward the plane-wave result as the (LCAO) basis is
enlarged (...) an atomic basis with double zeta and polarization is sufficient and in some cases, even necessary to ensure quantitative agreement with the plane-wave calculation", the authors also note that "the convergence can be rather slow".
But this is not different or more strange (pun intended) than the fact that you have to check your k-point sampling, real space mesh cut-off and other numerical parameters for convergence, by increasing them to see if the results don't change. For instance, I have seen that for most Carbon-based system, SZP seems to be sufficient, but the same is not true for Copper.
As for the band gap or HOMO-LUMO gap, this point is well known and discussed extensively in the literature, it will lead us too far to repeat all that here, but in short it is, as Nordland mentioned, something to be aware of in the interpretation of your results. Maybe the true curves need to be shifted, the onset of the current may be higher, etc. One way to check can be to attempt the same calculation with e.g. the extended Huckel method. Provided suitable parameter sets can be found, often the gap is improved, although the method as such may have other sources of inaccuracy compared to DFT. But if both methods give similar results, or you can extrapolate trends between them, then you can trust your results much more.
