The conjecture that all higher-dimensional grids are stackable

Let $P_k$ be the path graph on $k$ vertices, let $d$ be a positive integer, and let $k_1,\ldots,k_d$ be positive integers. Write the Cartesian product grid as

$$\mathop{\square}_{i=1}^d P_{k_i}.$$

Call a graph stackable if, starting with one cup at each vertex, all cups can be moved onto any prescribed target vertex according to the cup-stacking rules. Grid stackability conjecture. For all $d$ and $k_1,\ldots,k_d$, the grid

$$\mathop{\square}_{i=1}^d P_{k_i}$$

is stackable. The paper establishes the two-dimensional case; stackability in arbitrary grid dimension remains open.