Globevnik–Stout conjecture on holomorphic extension from tangent complex lines

Let $\Omega\subset\subset D$ be convex domains with $C^2$ boundaries in $\mathbb{C}^n$, and let $f$ be continuous on $bD$. Globevnik–Stout conjecture. If $f$ extends continuously to a function holomorphic on $D\cap l$ for every complex line $l$ tangent to $b\Omega$, then $f$ extends holomorphically to $D$. This conjecture concerns characterizing boundary values of holomorphic functions by their extensions along tangent complex lines; the source states that a partial solution is known, while the full assertion is presented as an open question.