= Solution
Write $N(a_1,\ldots,a_m)$ for the limiting <spreading model> norm. We show that deleting any one coefficient does not increase it; iteration then gives every finite suppression.
Fix an index $r$, the coefficients, and a small $\eta>0$. Choose a generation threshold so that all the relevant full and shortened combinations beyond it approximate their model norms within $\eta$. Fix the indices before position $r$ beyond this threshold. Since $(x_n)$ is a <weakly null sequence>, the <Mazur lemma> gives a finite convex combination
$$
z=\sum_{j\in F}\lambda_jx_j,\qquad\lambda_j\geq0,\quad\sum\lambda_j=1,\quad\|z\|<\eta,
$$
with every $j$ after the fixed prefix. Choose the remaining indices after $\max F$. For every $j\in F$, the full ordered combination with $x_j$ in slot $r$ has norm at most $N(a)+\eta$. Averaging these combinations gives
$$
\left\|\sum_{i\ne r}a_ix_{n_i}+a_rz\right\|\leq N(a)+\eta.
$$
The shortened combination has norm at most $N(a)+(1+|a_r|)\eta$, and approximates its own model norm within $\eta$. Letting $\eta\downarrow0$ proves
$$
\boxed{\left\|\sum_{i\in A}a_ie_i\right\|\leq\left\|\sum_i a_ie_i\right\|}
$$
for every finite set $A$. Thus the model is a <suppression-unconditional basic sequence> with constant one. The convex-combination argument allows the omitted coordinate to sit between two retained coordinates; simply letting a single omitted index tend to infinity with all others fixed would not respect their order.
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