A combinatorial question for divergent sequences

Suppose $$C=\{ a_{i,j} \}_{i,j \in \Bbb{Z}}$$ is a symmetric complex bisequence (which means $$a_{i,j}=a_{j,i}$$) such that $$\sum_{i \le j} \lvert a_{i,j} \rvert$$ diverges. Call $$P \subseteq \Bbb{Z}^2$$ square-symmetric if $$P=F \times F$$ for some finite set $$F \subseteq \Bbb{Z}$$. Denote by $$S$$ the collection of $$\sum_{(i,j) \in P} a_{i,j}$$ for all square-symmetric $$P$$'s. Must $$S$$ be unbounded? The square-symmetric restriction really annoyed me and I don't know where to start. Even a result about the case $$a_{i,j}=\pm 1$$ is of great help to me!

PS: The problem can be stated in a matrix-manner: Suppose the entry-wise $$l^1$$-norm of a countably-infinite symmetric complex matrix is unbounded, must the entry-sums of its finite principal minors be also unbounded?

• Counterexample: $a_{i,j}=1/\max(i^2,j^2)$. Apr 30, 2022 at 17:47
• @MikeEarnest This can not be counterexample since your numbers are positive. In this (and similar) case, the square-symmetric sums (indexed by $n$) correspond to $\{ (i,j)|1 \le i,j \le n \}$ are unbounded. Apr 30, 2022 at 17:55
• $\{(i,j)\mid 1\le i\le n \text{ and }1\le j\le n\}$ is not a finite union of $\{(k,k)\}$ and $\{(i,j),(j,i),(i,i),(j,j)\}$. Apr 30, 2022 at 17:57
• Never mind, I was confused, and thinking of "disjoint union" instead of union. Interesting question... Apr 30, 2022 at 18:14
• Is $P\subseteq \mathbb{Z}^2$ a "square-symmetric set" if and only if $P=F \times F$ for some finite set $F \subset \mathbb{Z}$? May 1, 2022 at 0:28