The multiplicity-sequence criterion for integral dependence

Let $I\subset J$ be arbitrary ideals in an equidimensional and universally catenary Noetherian local ring. The multiplicity sequence is the sequence of normalized leading coefficients of the bivariate Hilbert polynomial of the bigraded ring associated to $I$ and the maximal ideal. Multiplicity-sequence conjecture. The ideals $I$ and $J$ have the same integral closure if and only if they have the same multiplicity sequence. This conjecture is an analogue of Rees' multiplicity theorem and would provide a criterion for equality of integral closures without localization at prime ideals; the forward implication is known, while the converse is the conjectural part.