follows from [@Grundhofer]\*[Theorem 1.1]{}. As is well known, Tits proved that every thick irreducible spherical building of rank $\geq 3$ is Moufang (see [@Tits]\*[Addenda]{}), so that the conjecture above can be reduced to the following one, which appears as Question 1.4 in [@Grundhofer].

Let $\Delta$ be an infinite thick irreducible compact totally disconnected spherical building of rank $2$. If $\Delta$ is strongly transitive, then $\Delta$ is Moufang.

In the rest of this paper, we shall therefore focus on compact spherical buildings of rank $2$, also called **compact polygons**. More precisely, a compact polygon of diameter $m$ is called a **compact $m$-gon**. It is irreducible if and only if $m \geq 3$.

Our first main result may be viewed as a first step towards the conjecture above.

\[theorem:Characterization\] Let $\Delta$ be an infinite thick compact totally disconnected $m$-gon with $m \geq 3$ and let $G$ be a closed subgroup of the group $\operatorname{\mathrm{Auttop}}(\Delta)$ of topological automorphisms which is strongly transitive on $\Delta$. The following assertions are equivalent.

(i) $\Delta$ is Moufang.

(ii) For each panel $\pi$ of $\Delta$, the closure in $\operatorname{\mathrm{Homeo}}(\operatorname{\mathrm{Cham}}(\pi))$ of the group of projectivities $\Pi(\pi)$ acts as a convergence group on $\operatorname{\mathrm{Cham}}(\pi)$.

(iii) For each panel $\pi$ of 