Continuity of the projection function $proj$

Let $Erdahl(n)$ denote the space used in the paper, and for $f\in Erdahl(n)$ let $proj(f)=g$, where $(g,h)$ is the unique extremal pair associated with $f$. Continuity conjecture for $proj$. The function $proj$ is continuous. The preceding uniqueness result establishes that the projection is well defined and equivariant under the action of $\operatorname{AGL}_n({\mathbb Z})$, but the continuity assertion itself is presented without a resolution in the supplied text.