Hoffmann's conjecture for the first higher isotropy index

Let $\phi$ be an anisotropic quadratic form of dimension at least $2$ over a field. The first higher isotropy index is denoted by $\mathfrak{i}_{1}(\phi)$.

Hoffmann's conjecture. The integer $\mathfrak{i}_{1}(\phi)-1$ is the remainder of $\dim\phi-1$ modulo $2^s$ for some $s<\log_2(\dim\phi)$.

This conjecture describes the possible first higher isotropy indices of anisotropic quadratic forms and was originally stated with an additional characteristic restriction; the source expects it to remain valid in characteristic $2$, including for totally singular forms.