Vasconcelos' positivity conjecture for normal Hilbert coefficients

Let $(R,\mathfrak{m})$ be an analytically unramified Noetherian local ring of positive dimension, and let $I$ be an $\mathfrak{m}$-primary ideal. Vasconcelos' positivity conjecture. The first normal Hilbert coefficient satisfies

$$\overline{\mathrm{e}}_{1}(I)\geq 0.$$

The conjecture asserts nonnegativity of the first normal Hilbert coefficient for primary ideals in analytically unramified local rings.