Lorentzian affine hyperspheres with constant affine sectional curvature

We study affine hyperspheres M with constant sectional curvature (with respect to the affine metric h). A conjecture by M. Magid and P. Ryan states that every such affine hypersphere with nonzero Pick invariant is affinely equivalent to either (x(1)(2) +/- x(2)(2))(x(3)(2) +/- x(4)(2))...(x(2m-1)(2) +/- x(2m)(2)) = 1 or (x(1)(2) +/- x(2)(2))(x(3)(2) +/- x(4)(2))...(x(2m-1)(2) +/-x(2m)(2))x(2m+1) = 1 where the dimension n satisfies n = 2m - 1 or n = 2m. Up to now, this conjecture was proved if M is positive definite or if M is a 3-dimensional Lorentz space. In this paper, we give an affirmative answer to this conjecture for arbitrary dimensional Lorentzian affine hyperspheres.