Cayley graphs of the group$\mathbb Z^4$ that are limits of minimal vertex-primitive graphs of type $HA$

In the joint paper by Giudici, Li, Praeger, Seress, and Trofimov, it is proved that any graph that is a limit of vertex-primitive graphs of type $HA$ is isomorphic to a Cayley graph of the group $\mathbb Z^d$. Earlier, the author proved that for $d\le3$ the number of pairwise nonisomorphic Cayley graphs of the group $\mathbb Z^d$, which are limits of minimal vertex-primitive graphs of type $HA$, is finite (and obtained their explicit description). The present paper includes the construction of a countable family of such graphs for the case $d=4$; moreover, up to isomorphism there are only finitely many Cayley graphs of such a type outside this family.