The answer depends on the norm you are considering. The answer is no, already for $k=3$, for the operator norm (see the references in the question Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)?), but the answer is yes for the Hilbert-Schmidt norm (at least if you replace $O(\varepsilon)$ by $o(1)$, see the answers to the question).
Mikael de la Salle
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