A positive proportion of monic odd-degree hyperelliptic curves of genus g ≥ 4 have no unexpected quadratic points

Let Fg be the family of monic odd-degree hyperelliptic curves of genus g over Q. Poonen and Stoll have shown that for every g ≥ 3, a positive proportion of curves in Fg have no rational points except the point at infinity. In this note, we prove the analogue for quadratic points: for each g≥ 4, a positive proportion of curves in Fg have no points defined over quadratic extensions except those that arise by pulling back rational points from P¹.