Existence of beta extensions with full intertwining

Let $\theta$ be an $m$-realization of $\mathfrak{t}$ for $({\rm G},h)$ with parametrization $(({\rm V},h),\varphi,\Lambda,\beta)$. Let $\mathcal{C}$ be a chamber in the building of ${\rm G}_{\varphi(\beta)}$, let $\{\Lambda_j\}$ be the chosen representative lattice sequences, and let $\{\kappa_j\}$ be a family of beta extensions of $\{\theta_j=\tau_{\Lambda,\Lambda_j,\beta}(\theta)\}$. A family has full intertwining when ${\rm I}_{\rm G}(\kappa_i,\kappa_j)\supseteq {\rm G}_{\varphi(\beta)}$ for all $i,j$. Existence conjecture. There exists a family $\{\kappa_j\}$ of beta extensions with full intertwining. The existence of such families would provide the compatibility needed to reduce arguments to the depth-zero situation; the supplied text does not state whether this assertion is proved or remains open.