Positive p-rank for hyperelliptic curves with rational branch points

Assume $p\equiv 1\pmod 4$. Let $C$ be a hyperelliptic curve of genus at least $3$ whose branch points are all defined over $\mathbb{F}_p$. Rational-branch-point p-rank conjecture. The $p$-rank of $C$ is at least $1$. This conjecture is prompted by computational observations that found no $p$-rank-zero examples under these hypotheses; the paper gives no proof, so the claim remains open.