The flat-neck volume conjecture

Let $H^+\subseteq\mathbb{R}^{(a,b)}$ be the positive pseudosphere and let $N^+\subset H^+$ be a neck, with Riemannian volume $\operatorname{Vol}N^+$. Flat-neck volume conjecture. The volume $\operatorname{Vol}N^+$ is maximized when $N^+$ is flat.

This conjecture is presented as a related geometric assertion arising from the asymptotics of the norm-normalized polar tail probability. It remains open in the generality stated.