Fix $1\leq p<\infty$, and denote \begin{equation*}(\oplus_n\ell_p^n)_\infty=\left\{\left((a_i^{(n)})_{i=1}^n\right)_{n=1}^\infty:\left(\|(a_i^{(n)})_{i=1}^n\|_p\right)_{n=1}^\infty\in\ell_\infty\right\}\end{equation*} i.e. the $\ell_\infty$-sum of $\ell_p^n$'s, endowed with the obvious norm. Furthermore, if $X$ is a Banach space then we denote by $\ell_\infty(X)$ the $\ell_\infty$-sum of infinitely many copies of $X$.
Question. For which values $q\in[1,\infty]$ is $\ell_q$ isomorphic to a complemented subspace of $(\oplus_n\ell_p^n)_\infty$?
Let me summarize what I know so far. Clearly, $q=\infty$ works. Taking $q=p$ also works, as is mentioned in Remark 3.4 of the paper "On the mutually non isomorphic $\ell_p(\ell_q)$ spaces" by Cembranos/Mendoza.
One might conjecture that these are the only $q$'s that work, but actually whenever $p\in(1,\infty)$ we also get $\ell_2$ complemented. Here's how.
Theorem 1. $(\oplus_n\ell_p^n)_\infty$ is isomorphic to $\ell_\infty(\ell_p)$.
This is just from Remark 3.4 in Cembranos/Mendoza. Let us give some more info from that paper.
Corollary 1. The space $\ell_q$, $q\in[1,\infty]$, is isomorphic to a complemented subspace of $(\oplus_n\ell_1^n)_\infty$ if and only if $q=1$ or $q=\infty$.
(See Prop. 6.1.) On the other hand, Cembranos/Mendoza showed in Proposition 3.5 that $\ell_2$ is complemented in $\ell_\infty(\ell_p)$ whenever $1<p<\infty$. Hence:
Corollary 2. The space $\ell_2$ is isomorphic to a complemented subspace of $(\oplus_n\ell_p^n)_\infty$ if and only if $1<p<\infty$.
Using Theorem 1 above to replace "$\ell_\infty(\ell_p)$" with "$(\oplus_n\ell_p^n)_\infty$" in Proposition 4.3 of Cembranos/Mendoza, we also have the following.
Corollary 3. If the space $\ell_q$, $q\in(1,\infty)$, is isomorphic to a complemented subspace of $(\oplus_n\ell_p^n)_\infty$, $p\in(1,\infty)$, then either $1<p\leq q\leq 2$ or $2\leq q\leq p<\infty$.