The zeta-function conjecture

Let $\Omega$ be a lattice in $\mathbb{C}$ with algebraic invariants $g_2,g_3$. Let $p\in\mathbb{C}\smallsetminus\Omega$ with algebraic $\zeta(p)$, and let $\alpha$ be an algebraic number satisfying the stated exclusions. Zeta-function conjecture. If $\alpha\ne0,1,-1$, $\alpha^4\ne1$ when $g_3=0$, and $\alpha^6\ne1$ when $g_2=0$, then $\alpha p\notin\Omega$ and $\zeta(\alpha p)$ is transcendental. This is proposed as an analogue of Schneider's corollary for the Weierstrass zeta function; the exclusions remove the exceptional elliptic-curve automorphisms. The source gives no resolution, so the conjecture remains open.