The affine-independence conjecture for holomorphic extension on domains

Let $cOmega$ be a domain in $cmathbb C^n$ with a smooth boundary, and let $a_1,ccdots,a_{n+1}cincoverlinecOmega$ be $n+1$ points belonging to no complex hyperplane. Suppose that $fcin C(cpartialcOmega)$ extends holomorphically in each cross-section $LccapcOmega$.

Affine-independence conjecture. Then $fcin A(cpartialcOmega)$.

The conjecture proposes that $n+1$ affinely independent points suffice for the one-dimensional holomorphic extension property on a general domain, extending the preceding result for the complex ball. The source gives no resolution, so the conjecture remains open.