Brakensiek–Guruswami's approximate graph homomorphism hardness conjecture

Let $\mathbf{A}$ and $\mathbf{B}$ be non-bipartite loopless undirected graphs such that $\mathbf{A}\to\mathbf{B}$. The Brakensiek–Guruswami conjecture asserts that

$$\operatorname{PCSP}(\mathbf{A},\mathbf{B})$$

is $\operatorname{NP}$-hard. Here $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$ is the approximate graph homomorphism problem distinguishing instances homomorphic to $\mathbf{A}$ from instances not homomorphic to $\mathbf{B}$. This is presented as a stronger conjecture than the approximate graph-colouring conjecture; the source identifies graphs with loops or bipartite graphs as trivial or reducible to $2$-colouring, while the stated non-bipartite loopless case remains open.