Suppose that $A_1,\dots, A_k$ are matrices that any two of them can be approximated by commuting matrices. i.e. for any $i$ and $j$, there are matrices $A_i'$ and $A_j'$ such that $\|A_i-A_i'\|<\varepsilon$ and $\|A_j-A_j'\|<\varepsilon$ and $A_i'A_j'=A_j'A_i'$. Can we find matrices $X_1,\dots,X_k$ such that any two of them commute and $\|X_i-A_i\|<O(\varepsilon)$ for all $i$?

Here the matrix norm could be any unitary invariant norm. I'm specially interested in the operator norm and Hilert-Schmidt norm.