Not directly. High interest rates are actually directly inflationary because they increase the public sector’s deficit and every penny of that deficit is someon

Not directly. High interest rates are actually directly inflationary because they increase the public sector’s deficit and every penny of that deficit is someone in the private sector’s income. However the larger economy is a complex dynamic system. In order for an interest rate reduce price inflation or induce price deflation, it would have to reduce the ratio of money-looking-to-buy:goods-services-looking-to-sell. While higher interest rates might indeed reduce the number of potential buyers and the amounts they are willing to pay, those higher interest rates will also increase the cost of production for any producer or service provider that employs any sort of debt financing and thus probably reduce that side of the ratio too. My guess is that overall it's a wash, modulo animal spirits or whatever you want to call the human element of the markets. Incidentally, hysteresis is hugely important, and often confounding for figuring out these kinds of effects.