Lattice isomorphic Banach lattices of polynomials

We study D\'iaz-Dineen's problem for regular homogeneous vector-valued polynomials. In particular, we prove that if $E^*$ and $F^*$ are lattice isomorphic with at least one having order continuous norm, then $\mathcal{P}^r(^n E; G^*)$ and $\mathcal{P}^r(^n F; G^*)$ are lattice isomorphic for every $n\in \N$ and every Banach lattice $G$. We also study the analogous problem for the classes of regular compact, regular weakly compact, orthogonally additive and regular nuclear polynomials.