The h-reducedness conjecture for the normalization quotient

Let $R$ be a commutative ring with no nilpotent elements, and let $\overline{R}$ denote its integral closure. The quotient $\overline{R}/R$ is called $h$-reduced when it has the property asserted below. The h-reducedness conjecture. The $R$-module $\overline{R}/R$ is $h$-reduced. This conjecture is attributed to Huneke and Swanson's book and is presented as an open problem; the paper mentions partial answers.