Convexity conjecture for thick generalized affine buildings

Let $X$ be an affine building modeled over $\mathbb{A}=\mathbb{A}(\mathrm{R},\Lambda,T)$, where $T$ is a proper translation subgroup of $\Lambda^n$. Let $A$ be an apartment identified with $\mathbb{A}$, and fix an origin and a fundamental chamber. Let $r$ be the retraction onto $A$ centered at the corresponding sector germ, and let $\rho$ be the retraction onto $A$ centered at the opposite chamber at infinity. Assume that $X$ is thick with respect to $\overline{W}T$. For a special vertex $x$ in $A$, convexity conjecture.

$$\rho\bigl(r^{-1}(\overline{W}.x)\bigr)=\operatorname{conv}(\overline{W}.x)\cap(x+T).$$

This conjecture seeks an analogue, for generalized affine buildings with a proper translation subgroup, of the convexity result established earlier for the preceding class of buildings. It predicts that the image under the retraction $\rho$ is exactly the convex hull of the $\overline{W}$-orbit of $x$, restricted to the coset $x+T$.