Frölicher spectral-sequence degeneration across Bogomolov–Guan deformation spaces

Determine whether the Frölicher spectral sequence degenerates at the first page for every complex manifold in each Bogomolov–Guan deformation class of type BG_n, and, if degeneration fails universally, characterize the locus in the Bogomolov–Guan deformation space where the degeneration page jumps.

Background

The paper establishes results about de Rham formality and Betti numbers, but these do not determine the behavior of the Frölicher spectral sequence, which depends on the complex structure rather than solely on the underlying smooth manifold. The authors state that the distinguished Bogomolov–Guan fourfold has degeneration at E_1 and that E_1-degeneration holds on an open subset of the deformation space.

The unresolved issue is whether this open subset is the entire deformation space. If not, the problem also asks for a description of the locus where the degeneration page increases, including its position in the period space and the geometric features responsible for such a jump.