Lower bound on k-rank approximation of matrix

Question

I am trying to find a lower bound on $\sum_{l=1}^{m}\| A_l - W\Lambda_lW^T\|_F^2$ where $A \in S_{++}^{n \times n}$ with diagonal elements 1, $W \in R^{n \times k}$ and $\Lambda_l \in R^{k \times k}$ diagonal matrix having positive elements for a fixed $k$. $W$ and $\Lambda$ are the variables, and $A$ and $k$ are known. If $m = 1$, then I can use Eckart–Young–Mirsky theorem to get the lower bound, I don't know how to get the bound when $m > 1$. Any pointers would be really helpful.