Lower bound on the difference of singular values

Question

Suppose $X, U \in \mathbb{R}^{n \times r}$, $n>r$, with compact SVD $X = \Phi_1 \Sigma_1 \Theta_1$ and $U = \Phi_2 \Sigma_2 \Theta_2$, $\Phi_1,\Phi_2 \in \mathbb{R}^{n \times r}$ and $\Sigma_1,\Sigma_2,\Theta_1,\Theta_2 \in \mathbb{R}^{r \times r}$. Let $\mathfrak{R} = \{ \Psi \in \mathbb{R}^{r \times r}: \Psi \Psi^\top = \Psi^\top \Psi = I_r \}$ be the set of rotation matrices in dimension $r$. Suppose there exists some constant $c_1$ such that $$\min_{\Psi \in \mathfrak{R}} \|X-U \Psi\|_2 > c_1,$$ and principle angles between $X$ and $U$ are small, i.e. let $\Phi_1^\top \Phi_2 = A \Sigma B^\top \in \mathbb{R}^{r \times r}$ be the SVD, $A, \Sigma, B \in \mathbb{R}^{r \times r}$, we suppose $$\min_i \Sigma_{ii}>0.99.$$Then can we prove $$\max_i |(\Sigma_1 - \Sigma_2)_{ii}|>c_2$$or$$\max_i |(\Sigma_1^2 - \Sigma_2^2)_{ii}|>c_2$$where $c_2$ is a constant related with $c_1$.