No, It's The Other Way Around
Not only is Baker wrong about this, but the truth is the opposite of Baker's assertion. Because particularly fuel prices and to a lesser extent food prices (food commodities are as sensitive as energy commodities, but the commodity cost is particularly in America a much higher share of the retail price of fuel than it is of the retail price of food, not to mention the restaurant price of food) are largely determined by the prices set by global financial markets. This means first of all that they are more flexible than other prices and secondly it means that they are strongly affected by the dollar's exchange rate (the weaker the dollar, the higher will the dollar price of commodities be). And through it's monetary policy the Fed strongly influence the dollar's exchange rate.
All of this means that the Fed particularly in the short term have a very strong effect on food and fuel prices, indeed the short-term effect is often even greater than the long-term effect.
By contrast, most "core" prices are more or less "sticky", meaning that Fed policy at least in the short-term have little or no influence over them.
While there are other factors influencing food and fuel prices, such as the civil war turned international war in Libya because of a recent ill-advised decision, that doesn't change that the Fed can and does change them in whatever direction way they want, which in practice have tended to be upward. And besides, "core" prices are also influenced by other factors.