The parameter-count conjecture for even-dimensional combinatorial torus cube packings

Let $n$ be even, and consider combinatorial torus cube packings, where a packing is non-extensible if no further cube can be added while preserving the packing conditions. The number of parameters of a packing is denoted by $N(\mathcal{CP})$.

Even-dimensional parameter-count conjecture. There exist non-extensible combinatorial torus cube packings with $n+2$ cubes and

$$N(\mathcal{CP})=\frac{n(n+1)}{2}.$$

The preceding proposition proves the analogous minimal-cube and parameter statement for odd $n$, while this even-dimensional existence assertion is posed as an unproved conjecture in the supplied text.