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

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