Totaro's Question on Zero-Cycles on Torsors

Let $G$ be a smooth connected linear algebraic group and $X$ be a $G$-torsor. Totaro asked: if $X$ admits a zero-cycle of degree $d\ge 1$, then does $X$ have a closed 'etale point of degree dividing $d$? While the literature contains affirmative answers in some special cases, we give an example to show that the answer is negative in general.