Mináč–Tân's Kernel n-Unipotent Conjecture

Let $F$ be a field containing a primitive $p$-th root of unity, let $G=G_F(p)$ be the maximal pro-$p$ quotient of its absolute Galois group, and let $n\geq 3$ be an integer. A pro-$p$ group $G$ has the kernel $n$-unipotent property if

$$G_{(n)}=\bigcap\ker(\rho:G\to {\mathbb U}_n({\mathbb F}_p)),$$

where the intersection runs over all continuous homomorphisms and ${\mathbb U}_n({\mathbb F}_p)$ is the group of upper-triangular unipotent $n\times n$ matrices over ${\mathbb F}_p$. Kernel $n$-Unipotent Conjecture. The group $G=G_F(p)$ has the kernel $n$-unipotent property.