If a non-real logarithm λ and its conjugate are algebraic over one ring ℚ[x], e^{|λ|} is transcendental

Let $\lambda \in \mathbb{C}$ with $e^{\lambda}$ algebraic and $\operatorname{Im}\lambda \neq 0$. Suppose that $\lambda$ and $\bar\lambda$ are both algebraic over the ring $\mathbb{Q}[x]$, for some $x \in \mathbb{C}$. Then

is transcendental.

The hypothesis puts $\lambda$ and $\bar\lambda$ in an algebra of transcendence degree at most one, which is what four exponentials in transcendence degree one needs. With $x = \lambda$ this is DiazModulus.exp_abs_transcendental_of_conj_algebraic. With $\lambda = \log 2 + i\pi$ and $x = i\pi$ it gives DiazModulus.exp_abs_log_two_add_i_pi_transcendental.