Lifting Prym Divisors on a Bielliptic Loop of Loops

Let $C$ be a bielliptic curve over an algebraically closed nonarchimedean field of odd characteristic or characteristic zero, and let $f \colon \widetilde{C} \to C$ be an étale double cover. We prove that, if $f$ specializes to a certain double cover of a loop of loops, then every Prym divisor on th