On the relationship between $\mathrm{AK}$-stability and $\mathrm{BMO}$-regularity

Let $(X,Y)$ be a couple of Banach lattices of measurable functions on $\mathbb T\times\Omega$ having the Fatou property and satisfying a certin condition $(*)$ that makes it possible to consistently introduce the Hardy-type subspaces of $X$ and $Y$. We establish that the bounded $\mathrm{AK}$-stability property and the $\mathrm{BMO}$-regularity property are equivalent for such couples. If either lattice $XY'$ is Banach, or both lattices $X^2$ and $Y^2$ are Banach, or $Y=L_p$ with $p\in\{1,2,\infty\}$, then the $\mathrm{AK}$-stability property and the $\mathrm{BMO}$-regularity property are also equivalent for such couples $(X, Y)$.