  Let $p$ and $q$ be anisotropic quadratic forms over a field $F$ of
characteristic $\neq 2$, let $s$ be the unique non-negative integer such that
$2^s < \mathrm{dim}(p) \leq 2^{s+1}$, 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 $\mathrm{dim}(q)$ 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 $W(F)\rightarrow
W(F(p))$ has dimension divisible by $2^{s+1}$.
