Nonexistence of intrinsically aperiodic flats in compact triangle-square complexes

Let $K$ be a compact nonpositively curved triangle-square complex, let $\widetilde K$ be its universal cover, and let $F$ be an intrinsically aperiodic flat. Intrinsically aperiodic flats conjecture. If $K$ is a compact nonpositively curved triangle-square complex and $F$ is an intrinsically aperiodic flat, then $F$ does not embed into $\widetilde K$ and does not immerse into $K$. This conjecture would exclude the aperiodic radial flats that obstruct the proposed Gersten–Short geodesic approach to biautomaticity; the source gives no resolution.