Congruence lattices of free lattices in non-distributive varieties

We prove that for any free lattice F with at least $\aleph\_2$ generators in any non-distributive variety of lattices, there exists no sectionally complemented lattice L with congruence lattice isomorphic to the one of F. This solves a question formulated by Gr\"{a}tzer and Schmidt in 1962. This yields in turn further examples of simply constructed distributive semilattices that are not isomorphic to the semilattice of finitely generated two-sided ideals in any von Neumann regular ring.