Congruence Lifting of Diagrams of Finite Boolean Semilattices Requires Large Congruence Varieties

We construct a diagram D⊲⊳, indexed by a finite partially ordered set, of finite Boolean 〈∨, 0, 1〉-semilattices and 〈∨, 0, 1〉-embeddings, with top semilattice 2, such that for any variety V of algebras, if D⊲⊳ has a lifting, with respect to the congruence lattice functor, by algebras and homomorphisms in V, then there exists an algebra U in V such that the congruence lattice of U contains, as a 0,1-sublattice, the five-element modular nondistributive lattice M3. In particular, V has an algebra whose congruence lattice is neither joinnor meet-semidistributive. Using earlier work of K.A. Kearnes and Á. Szendrei, we also deduce that V has no nontrivial congruence lattice identity. In particular, there is no functor Φ from finite Boolean semilattices and 〈∨, 0, 1〉-embeddings to lattices and lattice embeddings such that the composition ConΦ is equivalent to the identity (where Con denotes the congruence lattice functor), thus solving negatively a problem raised by P. Pudlák in 1985 about the existence of a functorial solution of the Congruence Lattice Problem.