if it has been undergoing close-to-neutral selection, that implies that almost all possible mutations in that region are fitness-neutral.

There is no "neutral" evolution, as all DNA sequences are subject to several constraints, such as maintaining GC content and preventing promoters from popping out needlessly. There is also large variability of mutation rates along different DNA regions. Together, this results in high variance of "neutral" mutation rate, and because of huge genome, making it (probably) impossible to detect even regions having quarter of neutral mutation rate. I think this is the case here.

This extends what zslastsman written regarding structure.