Inflation's Impact on Asset Prices and Private Equity

It's anyone's guess what the future holds at this point, but what would be the effects on our financial system in the event that inflation returns? Intuitively, more inflation should mean higher asset prices. Yet, ultra-low interest rates (a bi-product of low inflation) have buoyed asset prices over the last decade, since, also intuitively, the PV of future cash flows increases when rates fall. The converse of this (higher interest rates as a result of higher inflation) would decrease your PV and thus pull down asset prices. E.g. the 1970s saw rampant inflation, and equities performed terribly.

Technological advances and an ageing population are inherently deflationary, but with less globalization, supply chain disruptions, and most importantly huge debt monetization and currency devaluation, I see at least one possible scenario where inflation may rear its head in the coming years. This is not to say that the Fed won't be able to deal with it adequately; it has a giant balance sheet and can easily sell assets/decrease the money supply, but given how hyper-fixated the market is on stimulus nowadays, it could easily throw a tantrum like it did in 2018 when Powell barely started quantitative tightening.

More food for thought: what would happen to private equity in an inflationary environment? Which would have a greater impact on long-term returns – depressed valuations (--> lower entry price --> higher theoretical returns) or a higher cost of capital (higher interest rates --> lower theoretical returns)? Does it even really matter given all of PE's structural industry tailwinds? (Apollo projects they'll triple their AUM over the next decade)