Nested solutions for products of powers of paths and complete graphs

Let $P_{n_1}$ denote the path on $n_1$ vertices and $K_{n_2}$ the complete graph on $n_2$ vertices. For nonnegative integers $d_1,d_2$ and integers $n_1,n_2\geq 2$, write $P_{n_1}^{d_1}\square K_{n_2}^{d_2}$ for their Cartesian product. A solution is a vertex ordering whose initial segments minimize the edge boundary, and the solutions are nested when these optimal initial segments can be chosen to form a chain under inclusion. Nested-solutions conjecture. If $d_1,d_2\geq 0$ and $n_1,n_2\geq 2$, then $P_{n_1}^{d_1}\square K_{n_2}^{d_2}$ has nested solutions. This would contribute to the general program of determining optimal orders for Cartesian products and, according to the source, would help settle Harper's question; the general two-dimensional case still requires special treatment and no general methods are known.