arc in $\mathbb D$ with endpoints coincident with those of $E_j^\ast$ and intersecting at an angle of $\theta_N$. Easy length-area/extremal length estimates in $T^N_J$ yield $\omega(z^N_J,\ell^N_{J,j},T^N_J)\lesssim e^{-\frac{c_0}{\delta_N}}$ for some constant $c_0$ not depending on $N$. Employing the maximum principle and following the mapping to the disc, we find that the length of the arcs $\gamma_j^\ast$ is $o(\frac1{\log N})$ by our assumptions on the sequence $\{\delta_n\}$. Now by elementary geometry, the disc determined by $\gamma_j^\ast$ has radius $r$ bounded by $$r\lesssim\frac{c_1}{\log N}e^{-\frac{c_0}{\delta_N}},$$ so that $r=o(1)$. The arcs $\gamma_j^\ast$, $1\le j\le 4$, are therefore (uniformly) hyperbolically separated, at least for $N$ greater than some $n_0$. Let us now define $D^N_{J,j}$ to be the simply connected domain which is the pre-image under $\phi^N_J$ of the domain bounded by $E_j^\ast\cup\gamma_j^\ast$. We then define domains $$\tilde T^{N-1}_I:=\hat T^{N-1}_I\cup\bigcup\{D^N_{J,j}:L^N_{J,j}\subset\hat T^{N-1}_I\}$$ for each multi-index $I$ of length $N-1$.

Proceeding inductively, let us suppose that the domains $\tilde T^n_J$ have already been constructed for $n>m\ge n_0$, and, fixing $(m+1,J)$, let $\phi^{m+1}_J$ be a conformal mapping from $\tilde T^{m+1}_J$ preserving the symmetries of the domain. Let $E_j=E_j(m+1,J)(\partial L^{m+1}_{J,j}\cap\partial\hat T^{m+1}_J)$, let $E_j^\ast=3E_j$, and let $\gamma_j^\ast$ be the circular arc in $\mathbb D$ with endpoints coincident with those of $E_j^\ast$ and 