14.38 (1999)

For every pro-$p$-group $G$ of $(2 \times 2)$-matrices for $p \neq 2$ an analogue of the Tits Alternative holds: either $G$ is soluble, or the variety of pro-$p$-groups generated by $G$ contains the group $\overline{\left\langle \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ t & 1 \end{pmatrix} \right\rangle} \leqslant SL_2(\mathbb{F}_p[[t]])$ (A. N. Zubkov, Algebra and Logic, 29, no. 4 (1990), 287–301). Is the same result true for matrices of size $\geqslant 3$ for $p \neq 2$?