Let $X$ be a smooth connected quasi-projective curve over $\mathbf{Q}$. Let $U$ be the pro-unipotent etale fundamental group of $X$ over $\mathbf{Q}_p$.

$U^1 = U$ and let $U^n =[U,U^{n-1}]$.

Let $n\geq 1$.

Why is $U^n/U^{n+1} \cong \mathbf{Q}_p(n)^{r_n}$ for some positive integer $r_n$?

I "know" this is true for $X=\mathbf{P}^1-\{0,1,\infty\}$ because M. Kim uses it in his article on Siegel's theorem and the motivic fundamental group.

Why is this true in general? (Here we should probably ask our curve to be hyperbolic.)

Is this true if we replace $X$ by a higher-dimensional variety in general?

Does the above property also hold for the "other" unipotent fundamental groups such as the pro-unipotent de Rham fundamental group of $X$? (Again, the answer is yes if $X$ is the projective line minus three points.)