  We discuss several questions which remained open in our joint work with M.
Sodin "Almost periodic Jacobi matrices with homogeneous spectrum,
infinite-dimensional Jacobi inversion, and Hardy spaces of
character--automorphic functions". In particular, we show that there exists a
non-homogeneous set $E$ such that the Direct Cauchy Theorem (DCT) holds in the
Widom domain $\bbC\setminus E$. On the other hand we demonstrate that the weak
homogeneity condition on $E$ (introduced recently by Poltoratski and Remling)
does not ensure that DCT holds in the corresponding Widom domain.
