  A Poincar\'e-Einstein metric $g$ is called non-degenerate if there are no
non-zero infinitesimal Einstein deformations of $g$, in Bianchi gauge, that lie
in $L^2$. We prove that a 4-dimensional Poincar\'e-Einstein metric is
non-degenerate if it satisfies a certain chiral curvature inequality. Write
$R_+$ for the part of the curvature operator of g which acts on self-dual
2-forms. We prove that if $R_+$ is negative definite then $g$ is
non-degenerate. This is a chiral generalisation of a result due to Biquard and
Lee, that a Poincar\'e-Einstein metric of negative sectional curvature is
non-degenerate
