I want to prove or find a counterexample that there exist constants $\mu>0, \rho>0$ such that the following inequality holds: \begin{align} (H + \mu M)^2 \succeq \rho M^2, \end{align} where $\mu>0, \rho>0$ are constants to be chosen, $H\in \mathbb{R}^{n\times n}$ is a fixed symmetric matrix with bounded eigenvalues $\lVert H\rVert\le \ell$, $M = I_n - Z \in \mathbb{R}^{n\times n}$ with $Z = \mathbb{1}_n\mathbb{1}_n^\top/n\in \mathbb{R}^{n\times n}$, $\mathbb{1}_n = (1, \dotsc, 1)^\top\in \mathbb{R}^{n}$, and $A \succeq B$ means that $A-B\succeq 0$ is a positive semidefinite matrix.

The matrix $M$ has some properties that might be helpful: (a) $M^2 = M$, (b) the eigenvalues of $M$ are $\lambda =1$ with multiplicity $n-1$ and $\lambda =0 $ with multiplicity $1$, (c) $M\mathbb{1}_n=0$.