Hoffmann–Laghribi's Pfister-neighbour conjecture for forms of small defect

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. A form is a Pfister neighbour if it is similar to a subform of a Pfister form of dimension less than twice the dimension of that Pfister form.

Hoffmann–Laghribi's conjecture. If $s<2r$ and $\varphi_1$ is defined over $F$, then $\varphi$ is a Pfister neighbour.

This conjecture rules out examples where the first higher anisotropic kernel descends to the base field without the original form being a Pfister neighbour. It is known when $s\leq4$, while the general case remains open.