Reciprocal-branch-point conjecture for the genus-two family J_2$\lambda,1$

Let $J_2(\lambda,1)$ be the genus-two family of translation surfaces in the stratum $\mathcal{H}(1,1)$, and let $a,b,c$ be complex parameters. A genus-two hyperelliptic curve is given by six branch points occurring in reciprocal pairs. Reciprocal-branch-point conjecture. The family of hyperelliptic curves corresponding to $J_2(\lambda,1)$ is given by

$$y^2=(x-a)(x-1/a)(x-b)(x-1/b)(x-c)(x-1/c).$$

The observed reciprocal pairing is supported by numerical experiments for rational $\lambda$ with $\mu=1$, whereas the phenomenon was not observed after changing $\mu$.