Finiteness conjecture for unramified cohomology of unirational varieties

Let $X$ be a unirational variety over an algebraically closed field of characteristic zero, and let $H^i_{nr}$ denote unramified cohomology. Unramified-cohomology finiteness conjecture. The unramified cohomology of $X$ is finite. Unramified cohomology is a birational obstruction to rationality, and the supplied text gives no resolution of this finiteness assertion.