Summary: Denote by \(F_5\) the \(3\)-uniform hypergraph on vertex set \(\{1, 2, 3, 4, 5\}\) with hyperedges \(\{123,124, 345\}\). J. Balogh et al. [Random Struct. Algorithms 48, No. 4, 641–654 (2016; Zbl 1341.05176)] proved that if \(p > K \log n /n\) for some large constant \(K\), then every maximum \(F_5\)-free subhypergraph of \(G^3(n,p)\) is tripartite with high probability, and showed that if \(p_0 = 0.1\sqrt{\log n} /n\), then with high probability there exists a maximum \(F_5\)-free subhypergraph of \(G^3(n, p_0)\) that is not tripartite. In this paper, we sharpen the upper bound to be best possible up to a constant factor. We prove that if \(p > C \sqrt{\log n} /n\) for some large constant \(C\), then every maximum \(F_5\)-free subhypergraph of \(G^3(n, p)\) is tripartite with high probability.