Given matrix $A \in \Bbb R^{n \times m}$ with $\frac{n}{m} = O(n^{\delta})$, where $0 < \delta < 1$, and "nearly orthonormal" rows, i.e., for any $a_i, a_j$ rows of $A$ with $i \neq j$:
$$\|a_i\| = 1, \quad |a_i^\top a_j| \leq \epsilon \tag{condition (1)}$$
where $\epsilon > 0$ is fixed. Define the column range of $A$ as:
$$\text{range}(A) = \{Ax: x \in \mathbb{R}^m\}.$$
Can we find matrix $B \in \Bbb R^{n \times m}$ such that $\text{range}(A) = \text{range}(B)$ but $\|b_i\| = 1, |b_i^\top b_j| > \epsilon$ for all $i \neq j$?