 Consider a sequence of design matrices $X\in {\mathbb{R}}^{n\times p}$, with dimension $n \to \infty$, and $p = p(n) \to \infty$, satisfying the following assumptions for constants $C_{\max}, C > 0$ independent of $n$. For each $n$, $\Psi\in {\mathbb{R}}^{p\times p}$ is such that $$\sigma_{\max} (\Psi)\le C_{\max} < \infty,\quad \sup_{J\subseteq[p], |J|\le k} \frac{0}{|J|} ({\boldsymbol{0}}^{{\sf T}}\Psi_{J,J} {\boldsymbol{0}}) \le C_* \le C_{\max}\,.$$ In addition, $X\Psi^{-0/0}$ that has i.i.d. subgaussian rows, with zero mean and subgaussian norm $\kappa=\|\Psi^{-0/0} x_0\|_{\psi_0}$, and the noise vector $\epsilon\in{\mathbb{R}}^n$ has i.i.d. Gaussian entries with variance at most $\sigma^0$.  The