As others have pointed out, Reich's comment “In sum, inflation isn’t driving most of these price increases. Corporate power is driving them" is kind of nonsense. Inflation does not drive price increases, inflation is price increases.
He blames corporate monopolistic industries for the price increases but disregards the low inflation we had for decades under those same monopolistic industries. Industries are capitalizing on the consumer's willingness to pay higher prices. The consumers might be a bit duped by the propaganda about huge growth in the money supply devaluing our currency leading to higher prices along with the complementary stories about shortages. Industry Monopolization just makes it easier for them to raise prices when the consumer is ready to accept them. With fewer competitors, they don't have to worry so much about some players not getting with the program.
Prices and profits will come down when we tighten our purse strings. This might happen because we wake up and send them a message that we won't be duped any longer or maybe when we don't have so much disposable income to spend as freely.