Star operations on overrings of Noetherian domains

Let R   be a Noetherian domain, and let Star(R)Star(R) denote the set of star operations on R. Call R star regular   if |Star(T)|≤|Star(R)||Star(T)|≤|Star(R)| for each overring T of R  . In the case where Star(R)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)Star(T) for each proper overring of a one-dimensional Noetherian domain R   implies finiteness of Star(R)Star(R).