Does the trickle down effect play a role in maintaining inequality?

Yes, absolutely. The "trickle down effect" maintains inequality by really being exactly the opposite of what it claims to be. Rich guy pays fewer taxes, and uses that money to get richer. No part of that equation includes the little guy.

Trickle down economics is why the number of billionaires in this country is growing exponentially while the poor are the poorest they've been in nearly 100 years.