On rational chain connectedness of globally +-regular varieties

We prove that globally $+$-regular varieties are rationally chain connected in dimension three and mixed characteristic with residue field characteristic $p>5$. We also introduce a notion of strongly globally $+$-regular, and show that varieties of arbitrary dimension which are strongly globally $+$-regular over a dense open subset of $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathbb{Z})$ are rationally chain connected.