Constant potentials do not minimise the fundamental gap on convex domains in hyperbolic space

We show that for every $n \geq 2$ and $D > 0$ there exist a convex domain $\Omega \subseteq \mathbb H^n$ with diameter $D$ and a convex potential $V$ on $\Omega$ such that the fundamental gap of the operator $-\Delta+V$ is strictly smaller than the fundamental gap of $-\Delta$. In comparison to previous work, this result requires more refined control of the eigenfunctions.