Extending homeomorphisms on Cantor cubes

We discuss the question of extending homeomorphism between closed subsets of the Cantor discontinuum $D^\tau$. For every set $P\subset D^\tau$ let $L_p$ be the set of cardinality $\lambda$ such that the $\lambda$-interior of $P$ is not empty. It is established that any homeomorphism $f$ between two proper closed subsets $P$ and $K$ of $D^\tau$ can be extended to an autohomeomorphism of $D^\tau$ provided the sets $L_p$ and $L_k$ do not have so many points of discontinuity and $f$ preserves the $\lambda$-interiors of $P$ and $K$.