= Chiral conformal-gauge BRST nilpotence is on shell
{title2=$s^2X=-c^+\partial_+c^-\partial_-X-c^-\partial_-c^+\partial_+X$}
In <conformal gauge>, let the odd <left-acting BRST differential> act by $sX=c^+\partial_+X+c^-\partial_-X$ and $sc^\pm=c^\pm\partial_\pm c^\pm$. The <graded Leibniz rule> gives the displayed expression for $s^2X$. It vanishes using the ghost <equations of motion> $\partial_-c^+=\partial_+c^-=0$, but it is not zero for arbitrary off-shell fields. For independent <Grassmann variables> $\xi,\rho$, take $c^+=\xi$, $c^-=\rho\sigma^+$ and $X=\sigma^-$; then $s^2X=-\xi\rho\ne0$. Each ghost itself satisfies $s^2c^\pm=0$ algebraically. With the eliminated-antighost transformation $sb_{\pm\pm}=i\theta_{\pm\pm}/\alpha'$ and $s\theta_{\pm\pm}=0$ on the solution space, the antighosts are nilpotent too. This on-shell closure is compatible with an off-shell total-derivative variation of the gauge-fixed action. A full off-shell <BRST quantization> uses the complete gauge transformation and, where required, auxiliary fields rather than silently dropping mixed ghost derivatives.
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