Hyperbolicity and near hyperbolicity of quadratic forms over function fields of quadrics

Let p and q be anisotropic quadratic forms over a field F of characteristic not equal 2, let s be the unique non-negative integer such that 2(s) < dim(p) <= 2(s+1), and let k denote the dimension of the anisotropic part of q after scalar extension to the function field F(p) of p. We conjecture that the dimension of q must lie within k of an integer multiple of 2(s+1). This can be viewed as a direct generalization of Hoffmann's Separation theorem. Among other cases, we prove that the conjecture is true if k < 2(s-1). When k = 0, this shows that any anisotropic form representing an element of the kernel of the natural restriction homomorphism W(F) -> W(F(p)) has dimension divisible by 2(s+1).