Tight closure

In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by .
Let formula_1 be a commutative noetherian ring containing a field of characteristic formula_2. Hence formula_3 is a prime number.
Let formula_4 be an ideal of formula_1. The tight closure of formula_4, denoted by formula_7, is another ideal of formula_1 containing formula_4. The ideal formula_7 is defined as follows.
Here formula_19 is used to denote the ideal of formula_1 generated by the formula_21'th powers of elements of formula_4, called the formula_23th Frobenius power of formula_4.
An ideal is called tightly closed if formula_25. A ring in which all ideals are tightly closed is called weakly formula_26-regular (for Frobenius regular). A previous major open question in tight closure is whether the operation of tight closure commutes with localization, and so there is the additional notion of formula_26-regular, which says that all ideals of the ring are still tightly closed in localizations of the ring.
 found a counterexample to the localization property of tight closure. However, there is still an open question of whether every weakly formula_26-regular ring is formula_26-regular. That is, if every ideal in a ring is tightly closed, is it true that every ideal in every localization of that ring is also tightly closed?