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 $\mathrm{\ne }2$, let $s$ be the unique non-negative integer such that $2^s<\mathrm{d}\mathrm{i}\mathrm{m}(p)\le 2^{s+1}$, 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 $\mathrm{d}\mathrm{i}\mathrm{m}(q)$ must lie within $k$ of a 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 $W(F)\rightarrow W(F(p))$hasdimensiondivisibleby$2^{s+1}$.