NP-hardness conjecture for promise CSPs between $\mathbf{LO}_k$ and $\mathbf{LO}_l$

For integers $2\leq k<l$, let $\mathbf{LO}_k$ and $\mathbf{LO}_l$ denote the corresponding symmetric relational templates, and let $\mathrm{PCSP}(\mathbf{LO}_k,\mathbf{LO}_l)$ be the promise problem whose instances homomorphically map to $\mathbf{LO}_k$ and whose outputs must homomorphically map to $\mathbf{LO}_l$. $\mathbf{LO}$-template hardness conjecture. For every $2\leq k<l$, $\mathrm{PCSP}(\mathbf{LO}_k,\mathbf{LO}_l)$ is NP-hard. The paper has classified the relevant three-element cases except for $\mathbf{LO}_3$; the smallest case $\mathrm{PCSP}(\mathbf{LO}_2,\mathbf{LO}_3)$ is therefore among the unresolved instances motivating the conjecture.