The alpha-constraint label-cover hardness conjecture

Let $G={\rangle (V,E),2R,\Psi\langle}$ be a label-cover instance whose constraints are alpha-constraints, where an alpha-constraint is defined by permutations $\pi_1,\pi_2:\{1,\ldots,2R\}\to\{1,\ldots,2R\}$ and the relation

$$\alpha=\{(2i-1,2i-1),(2i,2i-1),(2i-1,2i)\}_{i=1}^R.$$

For a labeling $L$, let ${\sf sat}(G)$ be the maximum fraction of constraints it satisfies, and let ${\sf isat}_t(G)$ be the largest relative size of a vertex set admitting a labeling by subsets of at most $t$ labels that satisfies all induced constraints. The alpha-constraint hardness conjecture. For every $\varepsilon>0$ and $t\in\mathbb{N}$ there exists some $R\in\mathbb{N}$ such that it is NP-hard to distinguish between

$${\sf sat}(G)=1$$

and

$${\sf isat}_t(G)<\varepsilon.$$

This conjecture is used for the reduction to approximate coloring with three colors and is described as seemingly stronger than Khot's conjectures; the source notes that the corresponding hardness for $t=1$ was implicitly proved previously, while the conjecture concerns every constant $t$.