= Solution
Part (d) makes the residual and therefore $\widehat\nu=X^T(Y-X\widehat\beta)$ unique, so $E$ is unique. The <Karush-Kuhn-Tucker conditions> from part (c) imply that every nonzero block satisfies $\lVert\widehat\nu^{(k)}\rVert_2=n\lambda\sqrt m$. Hence $\widehat\beta^{(k)}=0$ for $k\notin E$.
Any two minimizers have the same fitted value and vanish outside $E$. Their difference $h$ is therefore supported on $E$ and satisfies $\widetilde Xh_E=0$. If $\widetilde X$ has full column rank, then $h_E=0$, proving uniqueness of $\widehat\beta$.
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