Can the following conclusion hold?
There exist matrices $H^i\in\mathbb{R}^{m\times n}, (m\leq n, i\in\mathcal{N})$, and non-zero real numbers $\underline{h}^i$, and $\overline{h}^i$. If the following inequality holds \begin{align} {\underline{h}^i}^2 I_{m}\leq H^{i} (H^{i})^T \leq {\overline{h}^i}^2 I_{m} \end{align} and \begin{align} \sum_{i\in\mathcal{N}}(H^{i})^T H^{i} >0, \end{align} then \begin{align} \sum_{i\in\mathcal{N}}(H^{i})^T H^{i} \geq \min_{i\in\mathcal{N}}\{{\underline{h}^i}^2\} I_n \end{align}
Note: The expression matrix $M>0$ means matrix $M$ is positive definite. Similarly, the expression $A\geq B$ means $A-B$ is positive semi-definite.