Let $(a_{n,k}: n,k \ge 1)$ be an infinite matrix of positive reals such that $\sum_{k\ge 1}a_{n,k}=1$ for all $n\in \mathbf{N}$.
Question. Is it true that there exist a free ultrafilter $\mathscr{F}$ on $\mathbf{N}$ and a real sequence $(x_k: k\ge 1)$ such that $$\forall S\in \mathscr{F}, \quad \mathscr{F}\text{-}\lim_n \sum_{k\notin S}|a_{n,k}-x_k|=0?$$
Ps. If $(y_n: n\ge 1)$ is a real sequence and $\ell \in \mathbf{R}$, then $\mathscr{F}\text{-}\lim_n y_n=\ell$ means that $\{n \in \mathbf{N}: |y_n-\ell| < \varepsilon\} \in \mathscr{F}$ for all $\varepsilon>0$. This question is relation to a previous one here. However, the shown counterexample (the infinite identity matrix) does not work in this variant: indeed, it would be enough to set $x_k=0$ for all $k\in \mathbf{N}$, and $\mathscr{F}$ be any free ultrafilter.