Is all NP-hardness in the promise CSP explained by Datalog reductions?

There is an old tradition of putting prices on problems in mathematics, and in universal algebra, we used to offer alcohol. I remember one such price as a student, a few people contributed, and the final amount was four beers and a shot of vodka (or something like that). Building on this tradition, I decided to offer my own price of a bottle of whisky to a negative counterexample to one of my conjectures. This price was claimed sooner than expected, but I am happy with the attention it created, so I decided to announce a second Whisky problem at Dagstuhl in 2025.

The problem is about NP-hardness within the realm of promise CSPs. All the necessary definitions can be found in [¹]. Here is a formal statement:

problem statement

Does there exist a pair of finite structures $A$ and $B$ such that $\operatorname{PCSP}(A, B)$ admits a polynomial-time reduction from $\operatorname{CSP}(K_3)$ (and hence it is NP-complete), but it does not admit such a Datalog reduction?

Alternatively, we may ask whether there is a finite-template promise CSP that is NP-complete under log-space reductions, but it does not admit an existential positive reduction from 3-colouring. I am offering a bottle of a fine single-malt Scotch whisky for an example of any promise template (together with the proofs of all claims), that answers the above question positively. Either of the variants is acceptable, but only one bottle will be awarded.

Although I would prefer a human solution, and I hope that machines won’t be tempted by a bottle of whisky, I am willing to accept AI assited proofs as long as the use of AI is sufficiently explained, and a human takes responsibility for the proof. I reserve the right to withdraw the price if the proof is discovered with heavy AI assistance.