  We prove that there are no semi-finite generalized hexagons with $q + 1$
points on each line containing the known generalized hexagons of order $q$ as
full subgeometries when $q$ is equal to $3$ or $4$, thus contributing to the
existence problem of semi-finite generalized polygons posed by Tits. The case
when $q$ is equal to $2$ was treated by us in an earlier work, for which we
give an alternate proof. For the split Cayley hexagon of order $4$ we obtain
the stronger result that it cannot be contained as a proper full subgeometry in
any generalized hexagon.
