A numerically flat rank-two bundle without a holomorphic connection on a $\partial\bar\partial$-threefold

We construct a numerically flat holomorphic vector bundle of rank two on a compact complex threefold satisfying the ordinary $\partial\bar\partial$-lemma and prove that it admits no holomorphic connection, thereby giving,in particular, a negative answer to a question posed by Cao--Deng--Matsumura.