Logarithms algebraic over ℚ(π) with rational squared modulus are rationally proportional, up to conjugation

Let $u, v$ be non-zero complex numbers such that $e^{u}$ and $e^{v}$ are algebraic, $u$ and $v$ are algebraic over $\mathbb{Q}(\pi)$, and $|u|^{2} = u\bar u$ and $|v|^{2} = v\bar v$ are rational. Then there is $q \in \mathbb{Q}$ with

A non-zero logarithm $u$ of an algebraic number with $|u|$ algebraic would be a counterexample to Diaz's conjecture. The theorem says that the counterexamples algebraic over $\mathbb{Q}(\pi)$ with a rational squared modulus form at most one orbit under rational scaling and complex conjugation.

Proof. Since $v \neq 0$, the rational $|v|^{2}$ is non-zero, so $u\bar u = c\, v\bar v$ with $c = |u|^{2}/|v|^{2} \in \mathbb{Q}$. The conjugates are algebraic over $\mathbb{Q}(\pi)$ too, as $\bar u = |u|^{2}/u$ and $\bar v = |v|^{2}/v$. So $u, v, \bar u, \bar v$ generate a field of transcendence degree at most one (Transcendence.trdeg_adjoin_le_one_of_isAlgebraic_adjoin), and DiazModulus.log_pair_rigid_of_trdeg_one gives the rational $q$.

Novelty. Not found in the sources read (among them Diaz 2004 and 2007, Roy–Waldschmidt 1995 and 1997, and Waldschmidt 1973, 1974 and 2005). It follows in a few lines from the four exponentials theorem in transcendence degree one (M. Waldschmidt, Solution du huitième problème de Schneider, J. Number Theory 5 (1973), Corollaire 4; Theorem 1 of D. Roy and M. Waldschmidt, Proc. Japan Acad. 71 (1995)), which is DiazModulus.log_pair_rigid_of_trdeg_one on this site. The nearest statement found has the same shape and a different content: by the Gelfond–Schneider theorem, the non-zero algebraic $\beta$ with $e^{\beta}$ algebraic form either the empty set or a single orbit $\mathbb{Q}^{\times}\beta_{0}$ (G. Diaz, J. Théor. Nombres Bordeaux 16 (2004), Théorème 5).