Waldschmidt's conjecture on effective complexity bounds for numeration expansions

Let $G$ and $H$ be expansions whose corresponding recurrence relations have multiplicatively independent dominant roots. Let $H_G(n)$ and $H_H(n)$ denote the relevant expansion complexities, and let $M$ be a nonnegative integer. Waldschmidt's conjecture. There exists an effectively computable constant $\tilde C$, depending on $G$ and $H$, such that

$$H_G(n)+H_H(n)\leq M$$

implies

$$\log n\leq \tilde C^M.$$

This is presented as a stronger conjectural version of the theorem on numeration expansions, motivated by Waldschmidt's conjectured sharp lower bounds for linear forms in logarithms. The claimed bound would improve the powers of $\log n$ obtained from the currently best known Baker–Wüstholz estimates.