A Bernstein property of some affine Kähler scalar flat graph (Q2655720)

For a smooth, strictly convex solution \(u\) of the Abreu equation \[ \sum U^{ij}\left(\left[\det (u_{kl})\right]^{-1}\right)_{ij}=0 \] on \(\mathbb R^n\) (\(2\leq n\leq 4\)), where \((U^{ij})\) denotes the cofactor matrix of the matrix \((u_{ij})\), a Bernstein property, meaning that \(u\) is a quadratic polynomial if the norm of affine Kähler-Ricci curvature is bounded from above, is given.