"Tenía miedo de que su fama de secretismo acabara imponiéndose."
"He was afraid that his reputation for secrecy..."??? Can someone help me finish this translation? It is from a book I am reading. I think "acabara" is "to have just" done something. "Imponiédose" is "imposing it". The two together, however, don't make sense to me in this sentence. Even the use of the gerund form here is confusing to me.
Ayudame, por favor.