Here again, one could even replace strong transitivity by chamber transitivity in this conjecture. Historically, a similar problem had been suggested by J. Tits in the 1970’s for finite buildings. More precisely, Tits conjectured that a finite thick irreducible building of rank $\geq 2$ should be the building associated with a finite simple group of Lie type if and only if the building has a strongly transitive automorphism group, see [@Tits]\*[Conjecture 11.5.1]{}. Tits’ hope was that this could be helpful to the classification of the finite simple groups, which was an ongoing project at the time. Ironically, the latter classification was achieved first, and then used by Buekenhout and Van Maldeghem in their proof that Tits’ conjecture is indeed accurate, see [@Buekenhout].

As mentioned above, the conjecture is certainly true for connected buildings by the work of Burns and Spatzier; we may therefore assume that $\Delta$ is totally disconnected, since any disconnected compact building is so. Moreover, Grundhöfer, Kramer, Van Maldeghem and Weiss have proved that, under the same hypotheses as the conjecture, the building $\Delta$ is the spherical building associated with a semisimple algebraic group over a non-discrete locally compact field if and only if $\Delta$ is Moufang: this 