The coincidence conjecture for the two-torsion line bundles on the Hessian surface

Let $[f]\in U$ be general, and let $Y$ be the associated surface. The surface $Y$ has two $2$-torsion line bundles: $[?]$ the line bundle $[?]$ defined by $K_Y=3H|_Y+\eta$, and $[?]$ the line bundle $\eta'$ corresponding to its natural non-trivial unramified double cover. Coincidence conjecture. For $[f]\in U$ general, one has

$$\eta=\eta'.$$

This would identify the two geometrically arising non-trivial $2$-torsion line bundles on $Y$; the source presents this as an intriguing question, and no resolution is supplied here.