I tried the median filter to the SDSS F-star spectra in Lee et al. 2011 and the AMBRE model templates. That works well in both samples. The broad Balmer lines and narrow other lines survive at the same time. Because the whitened spectra with the median filter look better than those with Fourier transformation, I would use the median filter as the first option.
 
The figures are the results of median filter whitening. Each panel follows one in the figures in PIPE2D-648. The width of median filter is 72 pixels, which is comparable to 100 nm at 600 nm. Before applying the median filter, the lines and outliers are masked by sigma-clipping and the pixels around H alpha (645 nm < λ < 665 nm) are also masked. The blue part (<~450 nm) is really worse due to many lines there.
 
Examples
an SDSS spectrum (Teff~7000K):

 
an AMBRE template (Teff~6000):

 
 
 
 
 

Low temperature stars have many absorption lines. These can affect the result of the median filter whitening. This was pointed out in the 2D DRP telecon.
 
The whitened spectra of two low-temperature stars (Teff = 5400K and 6000K) are shown. As pointed out, there are a lot of small absorption lines and due to this, median filters are underestimating continua. In this media filtering process, I do the sigma clipping around a pre-whitened spectrum, but these small absorption lines are not clipped.
Another filter, e.g., a filter tracing the first quartile, may improve it.
 
Teff=5400K (SDSS star)

 
Teff=6000K (SDSS star)

We can close this ticket because we have discussed about this in the telecon. After generating flux calibration vectors using these whitened spectra, we will revisit the whitening study to improve the accuracy.