The generalised Albert form conjecture

Let $F$ be a field of characteristic different from $2$. For $n\geqslant 2$, let $\varphi$ be a quadratic form over $F$ such that

$$\mathsf{\dim}(\varphi)=2^n+2^{n-1},\qquad [\varphi]\in\mathsf{I}^n(F).$$

A quadratic form $\varphi$ is a generalised Albert form of degree $n$ if $2^n<\mathsf{\dim}(\varphi)<2^{n+1}$ and there exist $\pi_1,\pi_2\in\mathsf{GP}_n(F)$ with

$$[\varphi]=[\pi_1]+[\pi_2].$$

Generalised Albert form conjecture. Every such form $\varphi$ is a generalised Albert form.

The conjecture is explicitly stated for $n=4$ in the cited work and is proposed here for all $n\geqslant2$ because no counterexamples are known.