Polynomiality of region counts for fixed gain-function trees

Let $n$ be fixed, let $a$ vary, and fix a tree of gain functions. Consider the regions of the arrangement $\mathcal{A}^{a,a}_{n-1}$ associated to that tree. Polynomiality conjecture. The number of such regions is a polynomial in $a$. This predicts a uniform polynomial count for each fixed gain-function tree, refining the Catalan enumeration of possible tree types; no resolution is given in the source.