Cardinality of intervals between successive Sc-varieties

Determine the cardinality of the intervals [V(Sc(a1⋯ak)), V(Sc(a1⋯ak+1))] for every integer k≥3, and in particular establish whether each such interval contains continuum many varieties.

Background

The paper proves that several intervals in the subvariety lattice associated with S7 have cardinality continuum, including the intervals [V(Sc(a1⋯ak)), Nk+1] for k>2 and, by further discussion, the intervals [Nk+1, Nk+2] for k≥3. These results describe substantial portions of the lattice of subvarieties of the variety generated by S7.

However, the authors do not determine the size of the distinct intervals lying directly between successive varieties V(Sc(a1⋯ak)) and V(Sc(a1⋯ak+1)) when k≥3. They indicate that continuum cardinality is plausible but leave it unresolved.