Finiteness conjecture for thoroughly crumpled planes

Let $K$ be a compact nonpositively curved triangle-square complex, and let a thoroughly crumpled plane mean the type of flat plane defined in the source. Thoroughly crumpled planes conjecture. For any compact nonpositively curved triangle-square complex $K$, only finitely many distinct thoroughly crumpled planes immerse into $K$. Consequently, there is a global value $k$ depending only on $K$ such that Gersten–Short paths in any flat inside $\widetilde K$ that start and end within $1$ unit of each other synchronously $k$-fellow travel. This is proposed as a step toward proving biautomaticity: the paper establishes the corresponding fellow-travelling statement for each potentially periodic flat individually, but not the required finiteness or uniform global bound.