Counting disjoint hypercubes in Fibonacci cubes

We provide explicit formulas for the maximum number qk(n) of disjoint subgraphs isomorphic to the k-dimensional hypercube in the n-dimensional Fibonacci cube Γn for small k, and prove that the limit of the ratio of such cubes to the number of vertices in Γn is 12k for arbitrary k. This settles a conjecture of Gravier, Mollard, Špacapan and Zemljič about the limiting behavior of qk(n).