Classification conjecture for forms of nondefective height 1

Let $\varphi$ be a quadratic form of nondefective height $1$, and let $s$ denote the parameter occurring in its decomposition as a form of type $(r,s)$. Assume that

$$s \leq \frac{\operatorname{dim}\!\;\varphi}{3}.$$

Classification conjecture. Then $\varphi$ is a Pfister neighbour.

This conjecture proposes a classification of forms of nondefective height $1$ in the indicated range. The preceding proposition establishes the dimension formula $\operatorname{dim}\!\;\varphi=2^{n+1}-s$ under the same numerical hypothesis, but the stronger conclusion that $\varphi$ is a Pfister neighbour remains the conjectural part.