Kirby's weak Schanuel conjecture for n = 2, with Schanuel's conjecture over the kernel for n = 1 (trdeg Q(π, a, eᵃ) ≥ 2 for a ∉ 2πiQ), implies Diaz's conjecture

The hypothesis hWSC is the case $n = 2$ of Kirby's weak Schanuel conjecture (Kirby 2018, Conj. 1.5): if $\operatorname{trdeg}_{\mathbb{Q}}\mathbb{Q}(a, b, e^a, e^b) < 2$, then $ma + nb \in 2\pi i\mathbb{Z}$ for some integers $(m, n) \neq (0, 0)$. The hypothesis hSCk is the case $n = 1$ of Schanuel's conjecture over the kernel $2\pi i\mathbb{Q}$: for every $a \notin 2\pi i\mathbb{Q}$, $\operatorname{trdeg}_{\mathbb{Q}}\mathbb{Q}(\pi, a, e^a) \geq 2$. (At $a = 1$ it contains the algebraic independence of $e$ and $\pi$, which is open.)

Under both hypotheses Diaz's modulus conjecture (DiazModulus.DiazModulusConjecture) holds.

Proof. By DiazModulus.diaz_iff_single_relation_of_weak_schanuel it suffices that $t^2 + \pi^2$ is transcendental for real $t \neq 0$ with $e^t$ algebraic. If it were algebraic, $t$ would be algebraic over $\mathbb{Q}(\pi)$ and $e^t$ over $\mathbb{Q}$, so $\operatorname{trdeg}_{\mathbb{Q}}\mathbb{Q}(\pi, t, e^t) \leq 1$; but $t$ is real and non-zero, so not in $\mathbb{Q}\cdot2\pi i$, and hSCk gives at least $2$.

Novelty. Not found in the sources read; it is short. It combines DiazModulus.diaz_iff_single_relation_of_weak_schanuel (R7, not found in the sources read) with the kernel form of Schanuel's conjecture, a standard step.