The elliptic Schanuel conjecture

Let $\Omega$ be a lattice in $\mathbb{C}$, let $k$ be the associated elliptic curve's endomorphism field, and let $g_2,g_3$ be its invariants. Let $p_1,\dots,p_n$ be $k$-linearly independent elements of $\mathbb{C}\smallsetminus\Omega$. Elliptic Schanuel conjecture. At least $2n$ of the $2+3n$ numbers

$$g_2,g_3,p_1,\dots,p_n,\wp(p_1),\dots,\wp(p_n),\zeta(p_1),\dots,\zeta(p_n)$$

are algebraically independent. This is the elliptic specialization of the split semi-elliptic conjecture and is open in general, although important special cases are known.