Generating-function square-root conjecture for Q-curvature defects

Let

$${\mathcal G}(r)=1+\sum_{N\geq1}(-1)^N\Lambda_{2N}{r^N\over N!(N-1)!},$$

where $\Lambda_{2N}=Q_{2N}-{\mathcal Q}_{2N}$. Let $v(r)$ be the generating function defined by the volume coefficients. Square-root conjecture.

$${\mathcal G}\left({r^2\over4}\right)=\sqrt{v(r)}.$$

This reformulates the recursive Q-curvature relations as a generating-function identity. Obstructions are explicitly noted in the surrounding discussion, and the formula remains conjectural in general.