It is hard to establish superiority (without heavy causality) because it is narrow. The Iranian north of the strait is highly mountainous and everywhere in the strait, especially the 'shipping lane' and easy striking distance from onshore attack vectors.
The main concern therefore is military causality of American marines and Naval fleet. This is a highly asymmetric warfare. They can lay smart mines from a fishing boat and that can potentially damage an aircraft carrier or definitely blow up an VLCCF which cost hundreds of millions with millions of dollars worth of cargo.
So it's back to they would have to put boots on the ground on those mountainous shorelines to protect them. I see.