Transitivity of the unipotent subgroup on opposite-ended apartments

Let $A^7=g\cdot A^7$ for some $g\in B_{i,\infty}^7$, where $A^7$ is the standard apartment and $U_i$ is the corresponding unipotent subgroup. Transitivity conjecture. For each such $g$, there exists some $u\in U_i$ such that

$$u\cdot A_g^7=A^7.$$

This would describe the action of $U_i$ on the apartments based at the relevant end and is used in the paper to derive product decompositions of $B_{i,\infty}^7$. The paper gives an attempted proof but identifies an unresolved inductive step, so the conjecture remains open.