Bhargava's conjecture on the lowest odd-degree point of hyperelliptic curves

Let $g$ be the genus of a hyperelliptic curve. Consider locally soluble hyperelliptic curves that have points of odd degree. Bhargava's conjecture. As the family is ordered in the sense of the paper, $100\%$ of these curves have an odd-degree point of lowest degree exactly $g$ or $g+1$, whichever of these two integers is odd. This predicts the smallest odd degree represented by such a curve in the large-genus limit.