The problem is open because conditional logics admit a far richer and more complex semantic landscape than classical propositional logic. The semantics typically involve possible worlds with selection functions or sphere systems, and the interactions between different axioms governing these structures produce non-trivial dependencies that resist the combinatorial techniques used in the Boolean setting. Unlike the finite Boolean case where Post could enumerate all clones algorithmically, the extensions of CK are not easily enumerated, the semantic classes are not always first-order definable, and standard duality or representation theorems powerful enough to turn semantic comparisons into algebraic lattice computations are not yet established for the full conditional setting. The doctrinal and categorical generalizations of completeness, as appearing in recent work on Godel completeness via type space functors, suggest new machinery may be needed but also that connecting these categorical tools to the syntactic extension problem requires non-trivial translation work.