Hoffmann's conjecture on the first Witt index of quadratic forms

Let $\varphi$ be an anisotropic quadratic form over a field $F$, let $\mathrm{dim}\varphi\geq 2$, let $\frak{i}_{1}(\varphi)$ denote its first Witt index, and for a nonzero integer $n$ let $v_{2}(n)$ denote the $2$-adic exponent of $n$. Hoffmann's conjecture.

$$\frak{i}_{1}(\varphi)\leq 2^{v_{2}(\mathrm{dim}\varphi-\frak{i}_{1}(\varphi))}.$$

Hoffmann's original conjecture in characteristic different from $2$ was proved by Karpenko; the source presents the displayed statement in the broader characteristic-$2$ setting, so its status and applicability should be checked against that extension.