The term “conquer” implies effort, be it in sport or in battle. Within the context of this letter, the effort would be against the temptation to be satisfied with a cooled-down love (and perhaps to tolerate the Nicolaitans). The Lord, with the promise of a reward, spurs his people to return to their initial love. That return requires effort.
7 He who has an ear, let him hear what the Spirit says to the churches. To the one who conquers I will grant to eat of the tree of life, which is in the paradise of God.’