No, it is not at all obvious. Read up on the papers written about QE after the 2008 crisis. Monetary base increases significantly but the money multiplier falls in tandem. The net effect is that the price level is not strongly impacted by this type of program, outside of pre-empting deflation by providing liquidity to counter a potential fall in economic activity due to seizing credit markets.

See also "banks don't lend out reserves." The linkage from "cash" (i.e. bank reserves) to the market price level of goods is not very strong.