Maximality of the volume sum in inequality (21) under bubbling (t → 1, |c| < 1, p ∈ Φ(RP^n))

Determine whether the sum of n-dimensional volumes appearing in inequality (21), arising from the Veronese embedding Φ: RP^n → S^{m(n)−1}, the fold map F_{H_{p,t}} across a spherical cap H_{p,t}, and the Möbius transformation T_{−c}, is maximized in the bubbling regime characterized by |c| < 1, p ∈ Φ(RP^n), and t → 1; specifically, ascertain whether this degenerate configuration (non-reflected part tending to Φ(RP^n) and reflected part tending to an embedded n-sphere) indeed yields the maximal value of Vol_n(T_{−c}(Φ(RP^n))) + Vol_n(T_{−c} ∘ R_H(Φ(RP^n))).

Background

In proving an upper bound for λ_2 via trial functions, the paper arrives at inequality (21), where the right-hand side involves the sum of volumes of certain transformed images of the Veronese-embedded RP^n. The conjectural strategy suggests that the sum should be maximized in a 'bubbling' limit as the fold map degenerates (t → 1) with the center c inside the unit ball and the cap center p lying on the image surface Φ(RP^n).

Establishing this maximality would align the analytical bound with the geometric degeneration that, in dimension two, achieves sharp values. The authors provide supporting estimates (Propositions 17 and 18) but explicitly note that they have not been able to prove the maximality claim.