Characteristic-two analogue of the height-two bilinear-form classification

Let $\mathfrak{b}$ be an anisotropic bilinear form over $F$. Let $\phi_{\mathfrak{b}}$ denote its associated quadratic form, let $h(\mathfrak{b})$ be its height, let $h_{\mathrm{qp}}(\phi_{\mathfrak{b}})$ be the quasi-Pfister height, and let $\deg(\mathfrak{b})=d$. Assume

$$h(\mathfrak{b})=h_{\mathrm{qp}}(\phi_{\mathfrak{b}})=2,\qquad \dim\mathfrak{b}=2^{d+1}.$$

Characteristic-two height-two conjecture. There exist a $(d-1)$-fold Pfister form $\pi$ and a $4$-dimensional form $\mathfrak{c}$ of nontrivial determinant such that

$$\mathfrak{b}=\pi\otimes\mathfrak{c}.$$

This is proposed as the characteristic-two counterpart of the first case in the expected classification of height-two quadratic forms. The source does not give evidence of a resolution.