Unless you can reduce inequality to zero - Harrison Bergeron style [1] - you can't fix this by reducing inequality. It's an ironclad law of mathematics that 20% of the population will be part of the bottom fifth.

[1] https://en.wikipedia.org/wiki/Harrison_Bergeron

I believe the level of inequality, and the level of social mobility (which is probably related to the magnitude of inequality), is more important than the existence of inequality.

Are you aware of any academic literature on the subject?

It's not my area, but putting some keywords in Google Scholar yielded some results pretty quickly. E.g. this paper

http://www.tandfonline.com/doi/abs/10.1080/13504850701720197

"We investigate the relationship between inequality and intergenerational mobility. Proxying fathers' earnings with using detailed occupational data, we find that sons who grew up in countries that were more unequal in the 1970s were less likely to have experienced social mobility by the late-1990s."