Vishik's dimension conjecture for anisotropic quadratic forms in powers of the fundamental ideal

Let $F$ be a field with characteristic different from $2$, let $I^n$ be the $n$-th power of the fundamental ideal in the Witt ring $W(F)$, and let $\phi$ be an anisotropic quadratic form whose Witt class lies in $I^n$. Assume $n\geq1$ and $\dim(\phi)<2^{n+1}$.

Vishik's conjecture. There is some $i\in\{1,2,\dots,n+1\}$ such that

$$\dim(\phi)=2^{n+1}-2^i.$$

This conjecture gives the remaining possible dimensions below $2^{n+1}$ after the Arason–Pfister theorem and Vishik's exclusions of the intervening range. The source reports that Vishik announced a proof, but that proof was not available there; its resolution should therefore be checked.