Let $n\in\mathbb{N}$ and $p,q\in(1,+\infty)$ with $p^{-1}+q^{-1}=1$. Consider isometric embedding between $\mathbb{C}$-Banach spaces $$ \rho:\ell_p^n\to\ell_\infty(S, \ell_1^n),x\mapsto(f\cdot x)_{f\in S} $$ where $S$ is a countable dense subset of the unit sphere in $\ell_q^n$. I would like to know if this embedding gives a contractively complemented copy of $\ell_p^n$ inside $\ell_\infty(\ell_1^n)$. I know that the answer is "no" for sufficiently big $n$ but I need the answer for $n=2$.
I suspect that the answer is "no" even for $n=2$, because for $\mathbb{R}$-Banach spaces there is a simple proof exploiting the fact that $\ell_\infty^2\cong_1\ell_1^2$.
