Arens extensions of disjointness preserving multilinear operators on Riesz spaces and Banach lattices

Let $E_1,\dots ,E_m$ be (non necessarily Archimedean) Riesz spaces, let $F$ be an Archimedean Riesz space and let $A\text{\hspace{0.17em}⁣}:E_1\times \cdots \times E_m\to F$ be a regular disjointness preserving $m$-linear operator. We prove that all Arens extensions of $A$ are disjointness preserving if either $A$ has finite lattice rank or the spaces are Banach lattices and $F^{\ast }$ has a Schauder basis consisting of disjointness preserving functionals.