Termination conjecture for the refined Disk Tree algorithm

Let $G$ be a graph with tree-length $tl(G)$, and let $(T_i,\phi_i)$, $H_i$, $S_i$, $B_k(x_i)$, and property $P_i$ be as in the $(k,\ell)$-Disk Tree algorithm. Suppose that $k=\ell=tl(G)$, and that whenever there is an option between removing $z$ or $y$ while iteratively removing vertices from $S_i\cap B_k(x_i)$ until $T_i$ satisfies $P_i$, the algorithm removes $z$ if $d(z,x_i)>d(y,x_i)$. Termination conjecture for the refined Disk Tree algorithm. Under these conditions, there exists an $r$ such that $H_r=G$. This would establish termination at the tree-length parameter and support the conjectured $2$-approximation refinement.