Star operations on overrings of Noetherian domains

Let R be a Noetherian domain, and let Star(R) denote the set of star operations on R. Call R star regular if |Star(T)| ≤ |Star(R)| for each overring T of R. In the case where Star(R) is finite we show that star regularity becomes a local property, and, further assuming that R is one-dimensional and local with infinite residue field, we prove that R is star regular. We also study the question of whether finiteness of Star(T) for each proper overring of a one-dimensional Noetherian domain R implies finiteness of Star(R).