Totaro's Question for Adjoint Groups of Types A_{1} and A_{2n}

Let $G$ be a smooth connected linear algebraic group over a field $k$, and let $X$ be a $G$-torsor. Totaro asked: if $X$ admits a zero-cycle of degree $d \geq 1$, then does $X$ have a closed \'etale point of degree dividing $d$? We give an affirmative answer for absolutely simple classical adjoint groups of types $A_{1}$ and $A_{2n}$ over fields of characteristic $\neq 2$.