Kim's Selmer-dimension conjecture for hyperbolic curves

Let $F=\mathbb{Q}$ and let $X$ be a hyperbolic curve. For the étale unipotent fundamental group $U^{\mathrm{et}}$ and its level-$n$ quotient $U_n^{\mathrm{et}}$, let $H^1_f(G_T,U_n^{\mathrm{et}})$ be the Selmer variety and let $U_n^{\mathrm{DR}}/F^0$ be the corresponding de Rham quotient. Kim's Selmer-dimension conjecture. For some $n$, possibly very large,

$${ \text{dim} } H^1_f(G_T,U_n^{\mathrm{et}})<{ \text{dim} }(U_n^{\mathrm{DR}}/F^0).$$

This dimension inequality is intended to force the image of the rational or integral points in the de Rham quotient into a proper Zariski-closed subset, yielding finiteness by a non-abelian analogue of Chabauty's method. The conjecture is presented as the key Diophantine finiteness hypothesis and remains open in the stated generality.