The algebraic independence conjecture for logarithms

Let $\mathcal{L}$ be the $\mathbb Q$-vector space of logarithms of algebraic numbers:

$$\mathcal{L}=\big\{\lambda\mid e^{\lambda}\in\overline{\mathbb Q}\big\}.$$

For $n\geq 2$, let ${\sf LD}_n^{\mathbb Q}$ denote the relation of $n$ elements of $\mathcal{L}$ being linearly dependent over $\mathbb Q$, and let ${\sf AD}^n$ denote the relation of their being algebraically dependent. The algebraic independence conjecture for logarithms. In $\mathcal{L}$,

$${\sf LD}_n^{\mathbb Q}={\sf AD}^n\qquad\text{for all }n\geq 2.$$

This conjecture says that every $\mathbb Q$-linearly independent subset of logarithms of algebraic numbers is algebraically independent. It contains Baker's theorem as a subcase, since ${\sf LD}_n^{\mathbb Q}\subseteq {\sf LD}_n^{\overline{\mathbb Q}}\subseteq {\sf AD}^n$, and is presented in the source as an important open problem in transcendental number theory.