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

Let $\mathrm{\Omega }$ be a lattice in $\mathbb{C}$ with algebraic invariants and complex multiplication, let $\mathcal{E}$ be the elliptic curve associated with $\mathrm{\Omega }$, and let $\mathrm{\wp }$ be the Weierstrass function relative to $\mathrm{\Omega }$. Set $k:=\mathrm{End}(\mathcal{E}){\otimes }_{\mathbb{Z}}\mathbb{Q}\mathrm{.}$ We prove that if $t_1,\dots ,t_s$ are $\mathbb{Q}$-linearly independent algebraic numbers and $p_1,\dots ,p_n$ are $k$-linearly independent algebraic numbers, then the $s+n$ numbers $${\mathrm{e}}^{t_1},\dots ,{\mathrm{e}}^{t_s},\mathrm{\wp }(p_1),\dots ,\mathrm{\wp }(p_n)$$ are algebraically independent over $\bar{\mathbb{Q}}$. The proof uses the Tannakian description of the Lie algebra of the unipotent radical of the $1$-motive associated with these points.