The versality conjecture for folding-in

Let $F$ be a construction, let $G$ be the folding-in construction associated with the specified modification, and let $B$ be a building. Suppose a map $F\rightarrow B$ is given.

Folding-in versality conjecture. Every such map $F\rightarrow B$ extends to a map $G\rightarrow B$, although the extension need not be unique.

The source notes that nonuniqueness in the interior pieces of $G$ creates convergence issues, so the statement may require modification. No resolution is supplied.