The soficity conjecture for finitely generated groups

A group is sofic when every finitely generated group can be homomorphically embedded into a metric ultraproduct of permutation groups. Let $\mathcal{S}=(S_n,\|\cdot\|_H)$, where $\|\sigma\|_H=\frac{1}{n}\|\operatorname{supp}(\sigma)\|$ is the normalized Hamming norm, and let $\mathcal{S}^*_{\mathrm{met}}$ be its metric ultraproduct. The soficity conjecture. Every group is sofic; equivalently, every finitely generated group can be homomorphically embedded into $\mathcal{S}^*_{\mathrm{met}}$. This is presented as the main open problem concerning metric ultraproducts and sofic groups.