In [7] a new infinite class $\mathbb G_n$, $n \geq 0$, of near
polygons was defined. The near $2n$-gon $\mathbb G_n$ has the property that it contains
$\mathbb G_{n-1}$ as a big geodetically closed sub near polygon. In this paper, we
determine all near $2n$-gons, $n \geq 4$, having $\mathbb G_{n-1}$ as a big
geodetically closed sub near $2(n-1)$-gon under the additional assumption that
every two points at distance 2 have at least two common neighbours. We will
prove that such a near $2n$-gon is isomorphic to either $\mathbb G_n$, $\mathbb G_{n-1}
\otimes \mathbb G_2$, or $\mathbb G_{n-1} \times L$ for some line $L$.