Khot's unique games conjecture

Let $k$ be a positive integer, and consider an instance of the two-variable linear system $2$-Lin-$k$ over $\mathbb{Z}_k$, with equations

$$\mathbf{x}_i=\mathbf{x}_j+\mathbf{b}_l\pmod{k}.$$

An assignment is a map $\mathbf{x}=\{\mathbf{x}_1,\ldots,\mathbf{x}_n\}$ with each variable taking a value in $\{0,\ldots,k-1\}$. Unique games conjecture. For every $0<\epsilon<\frac{1}{2}$, given an instance in which a $1-\epsilon$ fraction of equations can be satisfied, there exists a value of $k$ such that no polynomial-time algorithm can find a solution satisfying at least an $\epsilon$ fraction of the equations. This is a central hardness assumption in approximation complexity and underlies optimality results for problems such as MAX-CUT. The source does not provide evidence resolving the conjecture.