Vishik's and Hoffmann's Pfister-neighbour conjectures for nondegenerate forms

Let $\varphi$ be a nondegenerate anisotropic form, let $X$ be its associated variety, and let $\Lambda^U(X)$ denote the set of upper motives appearing in the relevant motivic decomposition. Let $n$ be such that $2^n<\dim\varphi\leq2^{n+1}$.

Vishik–Hoffmann conjectures. The following assertions should hold:

1. If $|\Lambda^U(X)|=2$, then $\varphi$ is a Pfister neighbour.
2. If $\varphi$ has maximal splitting and $\dim\varphi>2^n+2^{n-2}$, then $\varphi$ is a Pfister neighbour.

These are presented as essentially due to Vishik and Hoffmann, respectively. They connect motivic decomposition and maximal splitting to the Pfister-neighbour property; the source gives no resolution of either assertion.