My answer might be ten years late, but it has been proven (at least when the boundary is sufficiently smooth) by Xiaodong Wang in An integral formula in Kahler geometry with applications, Comm. Contemporary Math. 19 (2017), no. 5, 1650063. See Theorem D therein.

My answer might be ten years late, but a great advance toward a proof of this conjecture, considering a lower bound on the Hermitian mean curvature, has been made (at least when the boundary is sufficiently smooth) by Xiaodong Wang in An integral formula in Kahler geometry with applications, Comm. Contemporary Math. 19 (2017), no. 5, 1650063. See Theorem D therein.