Kao's conjecture on Hamiltonicity of Cartesian products with odd paths

Let $G$ be a graph with a $\{P_2,P_3\}$-factor, meaning a spanning subgraph whose components are copies of $P_2$ or $P_3$, and let $P_n$ be the path on $n$ vertices. Write $\Delta(G)$ for the maximum degree of $G$, and call $G\Box P_n$ balanced bipartite when its two bipartition classes have equal cardinality.

Kao's conjecture. If $n\geq 4\Delta(G)-2$ and $G\Box P_n$ is balanced bipartite, then $G\Box P_n$ is hamiltonian.

This conjecture concerns Hamilton cycles in Cartesian products of graphs with paths. The surrounding results establish Hamiltonicity under related hypotheses, while the conjecture specifically addresses the odd-path case in which the standard construction using $P_3\Box P_n$ does not apply; its resolution is not given in the supplied text.