The core-free finite tractability conjecture via height-1 clone homomorphisms

Let $\mathbb{A}$ be a finite relational structure, and let $\mathscr{A}$ be its polymorphism clone. Let $\mathbf{1}$ be the clone of projections on a two-element set, and let an h1 clone homomorphism mean a homomorphism preserving identities of height $1$. Core-free finite tractability conjecture. One of the following holds: $\mathscr{A}$ maps to $\mathbf{1}$ via an h1 clone homomorphism, in which case $\operatorname{CSP}(\mathbb{A})$ is NP-complete, or $\operatorname{CSP}(\mathbb{A})$ is solvable in polynomial time. This is the core-free reformulation of the finite tractability conjecture, enabled by the equivalence between the relevant clone-homomorphism, cyclic-operation, and Siggers-operation conditions.