The split semi-elliptic conjecture

Let $\Omega$ be a lattice in $\mathbb{C}$, let $k$ be the field of endomorphisms of the associated elliptic curve, and let $g_2,g_3$ be its invariants. Let $t_1,\dots,t_s$ be $\mathbb{Q}$-linearly independent complex numbers, and let $p_1,\dots,p_n$ be $k$-linearly independent elements of $\mathbb{C}\smallsetminus\Omega$. Define

$$K=\mathbb{Q}(t_1,\dots,t_s,\mathrm{e}^{t_1},\dots,\mathrm{e}^{t_s},g_2,g_3,p_1,\dots,p_n,\wp(p_1),\dots,\wp(p_n),\zeta(p_1),\dots,\zeta(p_n)).$$

Split semi-elliptic conjecture. The transcendence degree of $K$ is at least $s+2n$, unless $2\pi\mathrm{i}\mathbb{Q}\subseteq\mathbb{Q}t_1+\cdots+\mathbb{Q}t_s$ and $\Omega\subseteq kp_1+\cdots+kp_n$, in which case it is at least $s+2n-1$. This conjecture unifies exponential and elliptic Schanuel-type statements and is shown in the source to be equivalent to the elliptico--toric conjecture; it remains open.