The Pfister-neighbour conjecture for forms with small defect parameter

Let $F$ be a field of characteristic $2$, and let $\varphi$ be an anisotropic form of type $(r,s)$ with $r+s\leq 2^n$. Write $\varphi_1$ for its first higher anisotropic kernel.

Pfister-neighbour conjecture. If $s<\frac{2^{n+1}}{3}$ and $\varphi_1$ is defined over $F$, then $\varphi$ is a Pfister neighbour.

This strengthens the known sufficient condition $s<2r$ under the same dimension restriction. The condition $s\leq4$ is known through the corresponding Hoffmann–Laghribi result, but the displayed broader range remains open.