Three-element reduction conjecture for minor-preserving maps

Let ${\mathbf I}$ be the clone of all idempotent operations on $\{0,1,2\}$. Define

$${\mathbf C}_2 \coloneqq \operatorname{Pol}(\{0,1\};\{(0,1),(1,0)\}),$$

let ${\mathbf C}_3$ be the clone considered in the article, and define

$${\mathbf T}_3 \coloneqq \operatorname{Pol}(\{0,1,2\};<).$$

Three-element reduction conjecture. Every clone on three elements that does not admit a minor-preserving map from ${\mathbf I}$ has a minor-preserving map to one of the following three clones on $\{0,1,2\}$: ${\mathbf C}_2$, ${\mathbf C}_3$, or ${\mathbf T}_3$.

This would reduce the classification of idempotent clones on three elements up to minor equivalence to three target clones. Together with the known reduction of clones containing an operation with image of size at most two to clones on a two-element set, it would provide an important step toward the broader countability conjecture.