Generalized exponential polynomial conjecture for mixed additive-function equations

Let $n$ be a positive integer, let $\mathbb{F}\subset\mathbb{C}$ be a field, and let $p_{1},\ldots,p_{n},q_{1},\ldots,q_{n}$ be fixed positive integers satisfying conditions C(i)--C(iii). Let $f_{1},\ldots,f_{n},g_{1},\ldots,g_{n}\colon\mathbb{F}\to\mathbb{C}$ be additive functions satisfying the mixed equation from the paper. A function is a generalized exponential polynomial function of degree at most $n-1$ if it has the corresponding finite exponential-polynomial representation of that degree. Generalized exponential polynomial conjecture. Every function $f_i$ and $g_i$ is a generalized exponential polynomial function of degree at most $n-1$. In particular, if

$$f_i(x)=D_i(x),\qquad g_i(x)=\widetilde{D}_i(x)\qquad (x\in\mathbb{F}^{\times}),$$

for derivations $D_i$ and $\widetilde{D}_i$, then the orders of $D_i$ and $\widetilde{D}_i$ are at most $n-1$. This conjecture would describe the full solution class of the mixed additive-function equation and would, in particular, bound the orders of derivations occurring among its solutions.