Asymptotic first boundary value problem for holomorphic functions of several complex variables

In 1955, Lehto showed that, for every measurable function $\psi$ on the unit circle $\mathbb{T},$ there is function $f$ holomorphic in the unit disc $\mathbb D,$having$\psi$asradiallimita\mathrm{.}e\mathrm{.}on$\mathbb T.$ We consider an analogous boundary value problem, where the unit disc is replaced by a Stein domain on a complex manifold and radial approach to a boundary point $p$ is replaced by (asymptotically) total approach to $p\mathrm{.}$