Scully's generalized separation conjecture for quadratic forms

Let $F$ be an arbitrary field, let $p$ and $q$ be anisotropic quadratic forms over $F$ of dimensions at least $2$, and let $s$ be the unique integer such that

$$2^s<\dim p\leq 2^{s+1}.$$

Let $F(p)$ be the function field of the integral affine quadric defined by $p=0$, let $\mathfrak{i}_0(q_{F(p)})$ be the isotropy index of $q$ over $F(p)$, and set

$$k=\dim q-2\mathfrak{i}_0(q_{F(p)}).$$

Scully's conjecture. If $q_{F(p)}$ is isotropic, then

$$\dim q=a2^{s+1}+\epsilon$$

for some positive integer $a$ and integer $\epsilon\in[-k,k]$ such that $\epsilon\equiv k\pmod{2}$. This conjecturally strengthens the separation theorem by constraining the dimension of a form that becomes isotropic over the function field of another quadric. The stated result has since been proved: Hoffmann established it in characteristic different from $2$, and Hoffmann and Laghribi extended it to characteristic $2$.