Beilinson's arithmetic Hodge index conjecture for intermediate Jacobians

Let $X$ be the fourfold considered in the paper, let $J^2(X)$ be its intermediate Jacobian, and let ${\mathsf L}:J^2(X)\to J^3(X)$ be induced by intersection with an ample cycle. Define the primitive part by $J^2(X)_0=\ker({\mathsf L})$. Beilinson's arithmetic Hodge index conjecture. The Beilinson--Bloch height pairing on $J^2(X)_0$ is positive definite.

This conjecture concerns the sign of the height pairing on the primitive component after separating the Lefschetz part $J^2(X)={\mathsf L}J^1(X)\oplus J^2(X)_0$. The supplied text gives no resolution or further evidence for the conjecture.