$J$-closed finite collections of Hardy-type subspaces

Several proofs of the following statement are given: if $X^0,\dots,X^n$ are BMO-regular lattices on the circle and $x\in X^0\cap\dots\cap X^n$, then the distances from $x$ to the Hardy-type subspaces $X^j_A$ are roughly attained at one and the same element of $\bigcap_jX^j_A$.