Borderline Trudinger–Moser inequality

Prove or disprove whether the ball-mass Trudinger–Moser inequality and attainment result in Theorem 7 remain valid at the critical parameter ν = 4π.

Background

Theorem 7 establishes finiteness and attainment of the truncated exponential supremum for 0 < ν < 4π, while the constant 4π is sharp in the sense that the conclusion fails for ν > 4π. The endpoint ν = 4π is not settled. Resolving it would clarify whether critical exponential-growth zero-mass Chern–Simons–Schrödinger equations can be treated directly in the ball-mass energy space.