Intermediate modular-balance condition and normal harmonic maps

Determine whether there exist polarized Lie algebras $(\mathfrak h,V(H))$ that admit a complement $W$ with the vertical modular character satisfying $\kappa_W|_{V(H)}=0$ but admit no modularly balanced complement; if such polarized Lie algebras exist, determine whether harmonic morphisms into the corresponding sub-Riemannian Lie groups are normal harmonic maps.

Background

Theorem \ref{hh:thm:balancednormal} gives a sufficient condition for a conformal submersion to be both a harmonic morphism and a normal harmonic map: the target polarized Lie algebra must admit a modularly balanced complement, meaning that the full modular mismatch vanishes. Theorem \ref{hh:thm:obstruction} gives a weaker necessary condition for normal harmonicity: some complement must have vanishing vertical modular character on the polarization. The question asks whether this gap is genuine and, if so, whether harmonic morphisms for targets in that intermediate class are nevertheless normal harmonic maps.

References

Are there polarized Lie algebras $(#1 h,V(H))$ that admit a complement $$ with $|_{V(H)}=0$ but no modularly balanced complement? If so, are harmonic morphisms into the corresponding sub-Riemannian Lie groups normal harmonic maps?

— Harmonic morphisms of sub-Riemannian Lie groups  (2609.04299 - Golo et al., 3 Sep 2026) in Section 8, Subsection 8.1, Subsection “Two questions,” Question \ref{hh:q:gap}