Solve $S_{nm} \sum_{j}x_{nj}\operatorname*{sign}\left( \sum_{k}x_{kj} S_{km} \right) \geq 0$ for $x_{kj}$

Question

Let $S_{nm} \in \{\pm1\}^{N\times M}$ be a matrix of signs. Can we find real numbers $x_{kj} \in \mathbb{R}^{N\times K}$, such that:

$$S_{nm} \sum_{j}x_{nj}\operatorname*{sign}\left( \sum_{k}x_{kj} S_{km} \right) \geq 0$$

I am interested in a regime where $M \ll N, K$.