Soprunova's mean-value formula conjecture for exponential sums

Let $\Lambda$ be a finite subset of $\mathbb R^n$, and let $F_1,\dots,F_n$ be exponential sums with real frequencies whose Newton polyhedra $\Delta_1,\dots,\Delta_n$ form a developed collection. Let $G$ be another exponential sum with real frequencies. If $M_{\Omega}$ denotes the mean value of $G$ over the zeros of $F_1=\cdots=F_n=0$, let $\Delta=\Delta_1+\cdots+\Delta_n$, and for each vertex $\alpha$ of $\Delta$ let $C_\alpha$ be the constant term defined from the formal exponential series associated with $F_1\cdots F_n$ and $G$. Let $k_\alpha$ be the corresponding combinatorial coefficients.

Mean-value formula conjecture. The mean value is equal to

$$M_{\Omega}=\frac{1}{(-2\pi)^n}\sum_{\alpha}k_\alpha C_\alpha,$$

where the sum is over the vertices $\alpha$ of $\Delta$.

This is the real-frequency analogue of the formula known for rational frequencies. The conjecture concerns whether the explicit combinatorial and coefficient expression remains valid for the mean value over zeros of a system with arbitrary real frequencies.