Classification conjecture for the Exists Right-Factor Problem and bounded f-cores

Let a variety be a class of algebras defined by identities, let $Z$ be a fixed algebra, and consider the Exists Right-Factor Problem for homomorphism factorization with fixed algebra $Z$. A variety has bounded $f$-cores if there is a bound on the size of the $f$-cores arising for its finite algebras.

Classification conjecture. The Exists Right-Factor Problem with fixed algebra $Z$ is in polynomial time for a given variety if and only if the variety has bounded $f$-cores.

This conjecture proposes a complete classification of when the fixed-algebra Exists Right-Factor Problem is solvable in polynomial time. The preceding discussion notes relationships among three conditions concerning bounded $f$-cores, but does not establish either implication in full; the conjecture is therefore presented as an open speculation.