Weak Schanuel conjecture for logarithms of algebraic numbers

Prove that if $\lambda_1,\dots,\lambda_m$ are $\mathbb{Q}$-linearly independent complex numbers whose exponentials are algebraic, then $\lambda_1,\dots,\lambda_m$ are algebraically independent.

Background

The paper uses this conjecture to establish algebraic genericity of Gram matrices and unit shapes for non-Galois totally real cubic fields and associated totally imaginary sextic fields. Under the conjecture, these shapes avoid every proper algebraic subvariety defined over Q‾\overline{\mathbb{Q}}.

The conjecture is a special logarithmic algebraic-independence statement and is not proved in the paper. Removing its use would make the relevant interior-shape and transcendence results unconditional.

References

\begin{conjecture}[Weak Schanuel Conjecture; , Conjecture 39.3.1] Let $\lambda_1,\dots,\lambda_m$ be $Q$-linearly independent complex numbers such that the complex numbers $\alpha_i=e{\lambda_i}$ are algebraic numbers. Then $\lambda_1,\dots,\lambda_m$ are algebraically independent. \end{conjecture}

— On the Geometry and Shapes of Rank 2 Log Unit Lattices  (2608.24736 - Cruz et al., 25 Aug 2026) in Conjecture 5.1, Section 5.1

\begin{conjecture}[See the paragraph above Conjecture 39.4.1 of ] Let $\alpha_1,\alpha_2, \alpha_3$ and $\alpha_4$ be logarithms of algebraic numbers. Then $$\det\left(\begin{bmatrix} \alpha_1&\alpha_2\ \alpha_3&\alpha_4 \end{bmatrix}\right)=0$$ if and only if either the two rows or the two columns are linearly dependent over $Q$. \end{conjecture}

— On the Geometry and Shapes of Rank 2 Log Unit Lattices  (2608.24736 - Cruz et al., 25 Aug 2026) in Conjecture 6.1, Section 6.2