Fix $1\leqslant p<\infty$, and let $(e_n)_{n=1}^\infty$ be an unconditional Schauder basis with some special properties which I'll discuss below. I'm trying to prove the following conjecture:
Conjecture 1. If finite-dimensional copies of $(e_n)_{n=1}^\infty$ are uniformly complemented in $(\bigoplus_{n=1}^\infty\ell_\infty^n)_p$ then they are uniformly complemented in either $\ell_p$ or else in $c_0$.
This needs some unpacking. If $X$ is a Banach space, we say that a Schauder basis $(e_n)_{n=1}^\infty$ has finite-dimensional copies uniformly complemented in $X$ iff there is a constant $C\in[1,\infty)$ such that for every $N\in\mathbb{N}$ there exists continuous linear maps $A_N:\text{span}(e_n)_{n=1}^N\to X$ and $B_N:X\to\text{span}(e_n)_{n=1}^N$ with $\|A_N\|\leq C$ and $\|B_N\|\leq C$, and such that $B_NA_N$ is the identity operator on $\text{span}(e_n)_{n=1}^N$.
Here, $(e_n)_{n=1}^\infty$ is the unit vector basis for a Lorentz sequence space. The definition of Lorentz sequence space is given in the link, but it probably isn't important. Probably all we need to know is that $(e_n)_{n=1}^\infty$ is 1-unconditional and 1-symmetric.
Really, what I want to show is the following:
Conjecture 2. Finite-dimensional copies of $(e_n)_{n=1}^\infty$ are not uniformly complemented in $(\bigoplus_{n=1}^\infty\ell_\infty^n)_p$.
It's already known (via ultrapower arguments) that finite-dimensional copies of $(e_n)_{n=1}^\infty$ are not complemented in $\ell_p$ nor in $c_0$. So Conjecture 1 implies Conjecture 2.
I confess I'm a little stumped on where to even start. Are there any tools that I should be looking at?
Really I'm asking for strategy ideas. How have similar problems been tackled?
Thanks guys.
EDIT: I'm going to edit in my ideas as they occur.
First, let's do something simple. Without loss of generality assume each $A_Ne_k$ is normalized, and write $$ A_Ne_k=((a_{(i,n)}^{(N,k)})_{i=1}^n)_{n=1}^\infty $$ In case $p=1$, we have $$ \frac{1}{L}\sum_{k=1}^Nc_k^*w_k\leqslant\sum_{n=1}^\infty\max_{1\leq i\leq n}\left|\sum_{k=1}^Nc_ka_{(i,n)}^{(N,k)}\right|\leqslant L\sum_{k=1}^Nc_k^*w_k $$ for some constant $L\in[1,\infty)$ and all scalar sequences $c_1,\cdots,c_N$. Here, $(c_k^*)_{k=1}^N$ is the nonincreasing rearrangement of $(|c_k|)_{k=1}^N$, and $(w_k)_{k=1}^\infty$ is the weight sequence in the definition of the Lorentz sequence space.