How exactly PoS is not making the rich richer? I fail to see your point.
I already explained in POS you can play your hand only once (dumb your stake) after that you have no more power over the system. In POW the largest miners can keep playing the system indefinitely.
POS changes everything.
1. In proof of work miners can play the system forever. They can dump their BTC when it's high and buy large amounts of BHC and start mining BHC which will rise it's price as you can see from the chart in the OP and what just happened over the past couple days. Then they dump BHC at 2-3 times the price and buy back BTC when it's low and move back to mining BTC... and on and on it goes and in the future it will be even worse when there are many more Bitcoin forks.
2. In Proof of Stake you can dump your stake (your holdings) only once! Once that is done you are out of power over the system and you can't play the system forever like in proof of work.