The second-level classification conjecture for low-dimensional forms in $I^n$

Let $F$ be a field, let $I^n(F)$ denote the $n$th power of the fundamental ideal in the Witt ring of $F$, and let $u$-fold Pfister forms be tensor products of $u$ binary forms. An Albert form is a $6$-dimensional symmetric bilinear form of trivial discriminant. Let $n$ be an integer with $n\geq 2$, and let $\mathfrak{b}$ be an anisotropic symmetric bilinear form of dimension $2^n+2^{n-1}$ over $F$. The second-level classification conjecture. If $\mathfrak{b}$ represents an element of $I^n(F)$, then there are an $(n-2)$-fold Pfister form $\mathfrak{c}$ and an Albert form $\mathfrak{d}$ over $F$ such that

$$\mathfrak{b}\simeq\mathfrak{c}\otimes\mathfrak{d}.$$

This conjecture extends the known classification of forms at the first level and the characteristic-not-two results for dimensions $6$ and $12$; its status is not resolved in the supplied source context.