Hoffmann's maximal splitting conjecture for quadratic forms

Let $q$ be an anisotropic quadratic form over a field of characteristic different from $2$ such that

$$2^n+2^{n-2}<\mathrm{dim}\,q\leq 2^{n+1}$$

for some positive integer $n\geq 2$. A quadratic form has maximal splitting when, writing $\mathrm{dim}\,q=2^r+m$ with uniquely determined integers $r\geq 0$ and $m\in[1,2^r]$, its first Witt index satisfies $i_1(q)=m$.

Hoffmann's maximal splitting conjecture. If $q$ has maximal splitting, then $q$ is a Pfister neighbour.

The conjecture proposes that, in the indicated dimension range, maximal splitting characterizes Pfister neighbours. The surrounding discussion states that maximal splitting is known for Pfister neighbours, while this converse was formulated by Hoffmann and remains unresolved here.