Stability conjecture for exterior Fraser-Sargent surfaces

Let $k$ and $l$ be relatively prime integers with $k>l>0$, and let $EFS_{k,l}$ be the exterior Fraser-Sargent surface in $\mathbb E^4\setminus\mathring{\mathbb B}^4$ associated with these parameters. Its index is denoted by $\operatorname{Ind}(EFS_{k,l})$. Stability conjecture. For all such $k,l$,

$$\operatorname{Ind}(EFS_{k,l})=0,$$

so every surface $EFS_{k,l}$ is stable. The authors explain that they were unable to compute the index rigorously; the conjecture is supported by a computer-assisted argument and remains open.