One-point extensions of generalized hexagons and octagons

In this note, we prove the uniqueness of the one-point extension SS of a generalized hexagon of order 2 and prove the non-existence of such an extension SS of any other finite generalized hexagon of classical order, different from the one of order 2, and of the known finite generalized octagons provided the following property holds: for any three points xx, yy and zz of SS, the graph-theoretic distance from yy to zz in the derived generalized hexagon SxSx is the same as the distance from xx to zz in SySy.