Certain numerical characteristics of $KN$-lineals

We study the properties of certain numerical characteristics in a normed lattice that characterize its conjugate space. A typical result is as follows: let $X$ be a $K_\sigma N$-space or a $KB$-lineal. If every sequence $\{x_n\}\subset X$ of pairwise disjoint positive elements with norms not exceedings 1 we have
$$ \varliminf_{n\to\infty}\frac1n||x_1\vee x_2\vee\dots\vee x_n||=0, $$
then all the odd conjugate spaces $X^*, X^{***},\dots$ are $KB$-spaces.