A power-commutator presentation for the group of truncated polynomials under substitution

Given a prime $p$, the Nottingham group $G(p)$ is the group of formal power series $x+a_2x^2+a_3x^3+\cdots$, $a_i\in{\bf Z}/p{\bf Z}$, under substitution. For $n\geq 1$, we write $G_n(p)$ for the quotient of $G(p)$ by its normal subgroup $K_n=\{x+a_{n+1}x^{n+1}+a_{n+2}x^{n+2}+\cdots\,|\, a_i\in {\bf