Biautomaticity conjecture under the Levitt–McCammond flat conditions

Let $X$ be a compact locally CAT(0) triangle-square complex such that no intrinsically aperiodic flat embeds in $\widetilde{X}$ and there are only finitely many distinct thoroughly crumpled flats that embed in $\widetilde{X}$. Let a Gersten–Short geodesic be a geodesic of the type used in the source, and let a regular path system satisfy the generalized fellow-traveler property when it has the property defined in the paper. Biautomaticity conjecture under the flat conditions. There is a vertex $v$ of $X$ such that Gersten–Short geodesics starting and ending at points of the orbit $\pi_1(X)\cdot v$ form a regular path system satisfying the generalized fellow-traveler property. In particular, $\pi_1(X)$ is biautomatic. The conjecture is presented as a conditional route to biautomaticity, and the supplied status evidence says that the original Levitt–McCammond conjecture remains open.