Complex interpolation of couple (X, BMO) for $A_1$-regular lattices

Recent results of A. Lerner concerning certain properties of the Fefferman-Stein maximal function are applied to show that $(\BMO, X)_\theta = X^\theta$, $0 < \theta < 1$, for a Banach lattice $X$ of measurable functions on $\mathbb R^n$ satisfying the Fatou property such that $X$ has order continuous norm and the Hardy-Littlewood maximal operator $M$ is bounded in $(X^\alpha)'$ for some $0 < \alpha \leqslant 1$.