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Another attempt. Take $K_{n+1} $ and three paths $abc, ade, afg$, $a\in K_{n+1} $, $b, c, d, e, f, g\notin K_{n+1} $. Any maximal matching contains at most one of edges $ab, ad, af$, thus at least two from three edges $bc, de, fg$. Hence any two maximal matchings have a common edge.