Hyperellipticity conjecture for the families J_g$\lambda,1$

For $g\in\{3,4,5\}$, let $J_g(\lambda,1)$ denote the corresponding family of translation surfaces of genus $g$. A compact Riemann surface is hyperelliptic if it admits a degree-two map to the projective line. Hyperellipticity conjecture. The surface $J_g(\lambda,1)$ is hyperelliptic for $g=3,4,5$. Numerical evidence comes from vanishing even theta constants for genera $3$ and $4$, and from modular-form equations and additional vanishing theta constants for genus $5$; the statement remains conjectural.