The single-base representation conjecture for sums of exponential terms

Let $a_n,c_n,b,d\rtin\mathbb{R}$ be real numbers, with $n\in\mathbb{N}$. The expression $a_nb^n+c_nd^n$ is considered as a sum of two terms with possibly different bases.

Single-base representation conjecture. For every $a_n,b,c_n,d\in\mathbb{R}$ and $n\in\mathbb{N}$, there exist $u_n,{\Greekmath 010D}\in\mathbb{R}$ such that

$$a_nb^n+c_nd^n=u_n{\Greekmath 010D}^n,$$

for all $n\in\mathbb{N}$.

This conjecture is proposed as a structural statement intended to help describe the space of finite Taylor measures, whose terms have the form $a_n{\Greekmath 010D}^n/n!$. Its resolution is not given in the supplied text.