Bryan–Oberdieck–Pandharipande–Yin genericity conjecture for hyperelliptic curves on Abelian surfaces

Let $B$ be a polarized Abelian surface, let $C$ be a hyperelliptic curve, and let $f:C\to B$ be a degree-$1$ morphism with image $\bar{C}$ such that the polarization $[\bar{C}]$ is generic. Let $\iota:C\to C$ be the hyperelliptic involution. Bryan–Oberdieck–Pandharipande–Yin conjecture. If $B$ is generic among polarized Abelian surfaces, then the differential of $f$ is injective at the Weierstrass points of $C$, and there is no non-Weierstrass point $p$ such that $f(p)=f(\iota(p))$. This concerns the intersection behaviour of curves on Kummer surfaces and a related problem about hyperelliptic curves on Abelian surfaces; the source provides no resolution status.