Algebraic independence of the exponential and Weierstrass $\wp$-functions

Let $Ω$ be a lattice in $\mathbb{C}$ with algebraic invariants and complex multiplication, let $\mathcal{E}$ be the elliptic curve associated with $Ω$, and let $\wp$ be the Weierstrass function relative to $Ω$. Set $ k:=\operatorname{End}(\mathcal{E}) \otimes_{\mathbb{Z}}\mathbb{Q}.$ We prove that i