Asymptotic for the number of star operations on one-dimensional Noetherian domains

We study the set of star operations on local Noetherian domains $D$ of dimension $1$ such that the conductor $(D:T)$ (where $T$ is the integral closure of $D$) is equal to the maximal ideal of $D$. We reduce this problem to the study of a class of closure operations (more precisely, multiplicative operations) in a finite extension $k\subseteq B$, where $k$ is a field, and then we study how the cardinality of this set of closures vary as the size of $k$ varies while the structure of $B$ remains fixed.