Lower-bound conjecture for the orientable covers of Fraser-Sargent surfaces

Let $k$ and $l$ be relatively prime integers with $k>l>0$, and let $\widetilde{IFS}_{k,l}$ denote the orientable cover of the Fraser-Sargent free-boundary minimal surface $IFS_{k,l}$. Write $\operatorname{Ind}(\widetilde{IFS}_{k,l})$ for its index. Orientable-cover index conjecture.

$$\operatorname{Ind}(\widetilde{IFS}_{k,l})\geq 6k+2l-1.$$

The bound is motivated by numerical experiments and is stated as an improvement of an earlier lower bound; it remains open.