"The Fed" which Reich refers to here, is the Federal Reserve, which has been raising interest rates consistently in an effort to fight inflation. In November of 2022, when Reich originally made this claim, the Consumer Price Index rose only 0.1%, suggesting that inflation in the marketplace was slowing. Yet, the Federal Reserve still wanted to raise interest rates.
Reich (the former Secretary of Labor), along with a host of other experts, quickly "sounded the alarm" at the Federal Reserve's insistency to keep interest rates high. They claim that there are " recessionary risks inherent in the Fed's approach, which aims to rein in prices by tamping down economic demand, slowing hiring, and cutting workers' wages." Even Federal Reserve Chair Jerome Powell said "'pain' is on the horizon for families."
So, yes, even the head of the Federal Reserve has confirmed that the brunt of inflation's costs will be felt by American workers and their families.
Source: https://www.commondreams.org/news/2022/12/13/inflation-cools-experts-warn-powell-think-twice-hiking-interest-rates-again