The celebrated Osterwalder-Schrader (OS) reconstruction results provide conditions verified by Euclidean n-point correlation functions to produce a Wightman quantum field theory. This talk aims to show a conformal version of the OS axioms, including the linear growth condition, for n-point correlation functions defined for a reasonable class of unitary full Vertex Operator Algebras (VOAs). Such VOAs can be seen as extensions of commuting chiral and anti-chiral VOAs, introduced to describe compact 2D conformal field theories. The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page