It is well known that two $n\times n$ symmetric positive semidefinite matrices $A$, $B$ such that $AB=0$ are simultaneously diagonalizable.

My question is related to the existence of a specific simultaneous diagonalization in the following sense: Let $\{A_k\}$, $\{B_k\}$ be two sequences of symmetric matrices converging to positive semidefinite matrices $A$ and $B$, respectively, such that $AB=0$. Is it the case that there exist a basis $\{v_i^k\}$ of eigenvectors of $A_k$ and a basis $\{w_i^k\}$ of eigenvectors of $B_k$, for all k, such that each $v_i^k$ and $w_i^k$ converge to some $c_i$ such that $\{c_1,\dots, c_n\}$ form a simultaneous basis of eigenvectors for $A$ and $B$?