Idempotence of the birational pre-closure

Determine whether the birational pre-closure operation J ↦ J^{Bir}, defined for an ideal J in a reduced Noetherian ring R by J^{Bir} := ⋃_{Y→Spec R} ker(R → H_0(R/J ⊗^L RΓ(Y, O_Y)) where the union runs over all proper birational morphisms Y → Spec R, is idempotent; specifically, show whether (J^{Bir})^{Bir} = J^{Bir} holds for every ideal J ⊆ R.

Background

The paper introduces the birational pre-closure J^{Bir} of an ideal J in a reduced Noetherian ring R as an analogue of plus-closure in the birational direction. It is defined via a union over proper birational maps Y → Spec R of the kernels of the natural maps R → H_0(R/J ⊗^L RΓ(Y, O_Y)). The authors show that their main derived Briançon–Skoda containment implies \overline{J^{n+k-1}} ⊆ (J^k)^{Bir}, and they relate J^{Bir} to established closure operations in various characteristics, including plus closure and extended plus closure.

They also prove that, under resolution of singularities in characteristic zero or low dimension, J^{Bir} can be computed using any resolution. Despite these structural results, it remains unsettled whether J^{Bir} itself satisfies the defining property of a closure operation, namely idempotence. The authors explicitly pose this as an open question.