Lasso sign recovery from cone invertibility

Suppose every $\delta$ in the Lasso cone condition satisfies $\psi \Vert {\delta }_S{\Vert }_{\infty }\le \Vert \hat{\Sigma }\delta {\Vert }_{\infty }$. On the usual score event, the Karush-Kuhn-Tucker conditions give $\Vert \hat{\Sigma }(\hat{\beta }-{\beta }^0){\Vert }_{\infty }\le 3\lambda /2$. Hence every active-coordinate error is at most $3\lambda /(2\psi )$, and a larger minimum nonzero signal guarantees coordinatewise sign recovery.