Beraldo–Kanel-Belov–Kontsevich conjecture on the image over the infinite-prime base

Let $R$ be a commutative ring, let $R_{\infty}$ be the reduction modulo the infinite prime, let $i:R\rightarrow R_{\infty}$ be the tautological inclusion, and let

$$\phi_R:\operatorname{Aut} W_{n,R}\rightarrow \operatorname{Aut} P_{n,R_{\infty}}$$

be the group homomorphism described in the source. Beraldo–Kanel-Belov–Kontsevich conjecture. The image of $\phi_R$ belongs to

$$\operatorname{Aut} P_{n,i(R)\otimes \mathbb Q}.$$

This conjecture asserts that the reduction construction has image defined over the smaller coefficient algebra $i(R)\otimes\mathbb Q$, rather than only over $R_{\infty}$. The source attributes it to Conjecture 3 of the cited work, but gives no resolution evidence.