The type A_2 shift conjecture for supported edge chains

Let $\Delta$ be the Bruhat–Tits building, let $\mathbb{X}$ be the coefficient system, let $X_{0,2}$ be the specified set of edges, and let $I$ be the relevant subgroup. Let $C_1(\Delta,\mathbb{X})$ be the space of edge chains and let $\varepsilon_0$ be the boundary map. A chain is $(I,1)$-shift invariant when it has the shift-invariance property defined in the source. Shift-invariance conjecture. If $\beta\in C_1(\Delta,\mathbb{X})$ is non-zero only on edges in $X_{0,2}$, and $\varepsilon_0(\beta)$ is non-zero only on vertices in $\Delta_1$, then $\beta$ is $(I,1)$-shift invariant.

This conjecture is motivated by the preceding shift results, which show that shift invariance would allow one to move the chain into a smaller region. The supplied text does not give a resolution.