Generalised Alspach conjecture for decompositions into double-rays and Hamiltonian circles

Let $G$ be a $2k$-regular Cayley graph of an abelian group. For an integer $i$, say that $G$ satisfies condition $P_i$ when every finite cut $F$ with two infinite components satisfies

$$|F|\equiv i\pmod 2.$$

Generalised Alspach conjecture. If $G$ satisfies condition $P_i$, then $G$ has a decomposition into $i$ Hamiltonian double-rays and $k-i$ Hamiltonian circles. Here a Hamiltonian double-ray is a spanning two-way infinite path, while a Hamiltonian circle is the corresponding topological generalisation of a Hamiltonian cycle in an infinite graph. The parity condition is motivated by the fact that finite cuts with two infinite components must have parity matching the number of double-rays in such a decomposition; the paper establishes the claim in the $4$-regular case, while the stated general form remains open.