$\Delta$ be a thick compact $m$-gon with $m \geq 3$ and $C, C'$ be opposite chambers of $\Delta$. Denote by $D_0$ and $D_1$ the two chambers adjacent to (but different from) $C$ in the apartment containing $C$ and $C'$ and by $v_0$ (resp. $v_1$) the common vertex of $C$ and $D_0$ (resp. $D_1$). Let also $E_0$ (resp. $E_1$) be a chamber having vertex $v_0$ (resp. $v_1$) but different from $C$ and $D_0$ (resp. $D_1$). Let finally $U$ and $U'$ be closed subsets of $\operatorname{\mathrm{Cham}}\Delta$ such that every chamber of $U$ is opposite every chamber of $U'$. Then for all $\varepsilon > 0$, the (possibly empty) set $$J_\varepsilon(C,C',U,U',E_0,E_1) := \left\{ \varphi \in \operatorname{\mathrm{Auttop}}(\Delta) {\;\ifnum\currentgrouptype=16 \middle\fi|\;}\begin{array}{l}
\varphi(C) \in U, \ \varphi(C') \in U', \\
\rho(\varphi(C), \varphi(B_{v_i}(D_i, \varepsilon))) \geq \varepsilon  \ \ \forall i \in \{0,1\},\\
\rho(\varphi(D_i), \varphi(B_{v_i}(C, \varepsilon))) \geq \varepsilon  \ \ \forall i \in \{0,1\},\\
\rho(\varphi(C), \varphi(E_i)) \geq \varepsilon  \ \ \forall i \in \{0,1\},\\
\rho(\varphi(D_i), \varphi(E_i)) \geq \varepsilon  \ \ \forall i \in \{0,1\}\\
\end{array}\
\right\}$$ is compact, where $\rho$ is some fixed metric associated to the topology on $\operatorname{\mathrm{Cham}}\Delta$ and $B_{v}(C, r)$ denotes the open ball in $\operatorname{\mathrm{Cham}}(v)$ centered at $C$ and of radius $r$ (with respect to the 