Biautomaticity conjecture for compact nonpositively curved triangle-square complexes

Let $K$ be a compact nonpositively curved triangle-square complex, meaning a complex built from triangles and squares with nonpositive curvature, and let $\pi_1(K)$ denote its fundamental group. Biautomaticity conjecture. The fundamental group $\pi_1(K)$ is biautomatic. This would extend the Gersten–Short theorem from complexes consisting solely of triangles or solely of squares to complexes mixing both types of cells; the paper presents it as the main conjecture, with the proposed geodesic construction and its required fellow-travelling and regular-language properties still to be established.