Curvature estimates on a parabolic flow of Fei-Guo-Phong

In \cite{FGP}, Fei, Guo and Phong established a criteria for the long-time existence of their parabolic flow from $11$-dimensional supergravity, which involves Riemannian curvatures $\mathrm{R}\mathrm{m}(g(t))$ and 4-forms $F(t)$. In this paper, we obtain a new criteria for the long-time existence of the same flow, which involves only Ricci curvatures $\mathrm{R}\mathrm{i}\mathrm{c}(g(t))$, $F(t)$, but as well as ${\mathrm{\nabla }}_{g(t)}F(t)$.