Hodge-Tate prismatic crystals and Sen theory

Let $K$ be a mixed characteristic complete discrete valuation field with perfect residue field, and let $K_\infty/K$ be a Kummer tower extension by adjoining a compatible system of $p$-power roots of a chosen uniformizer. We use this Kummer tower to reconstruct Sen theory which classically is obtained using the cyclotomic tower. Using this Sen theory over the Kummer tower, we prove a conjecture of Min-Wang which predicts that Hodge-Tate prismatic crystals are determined by the Sen operator; this implies that the category of (rational) Hodge-Tate prismatic crystals is equivalent to the category of nearly Hodge-Tate representations.