Factorization and squarefreeness conjecture for dimer packings on odd square lattices

Let $a_N$ denote the number of dimer packings of the odd lattice $(2k+1) \times (2k+1)$, and let $c_k$ be the integer in the factorization below. Factorization and squarefreeness conjecture. For the odd lattice $(2k+1) \times (2k+1)$,

$$a_N=2^k c_k,$$

where $c_k$ is an odd integer. Furthermore, when $k>1$, $c_k$ is squarefree: its prime decomposition contains no repeated factors. The conjecture identifies an arithmetic pattern in the number of dimer packings on odd square lattices; the accompanying congruence pattern for $c_k$ is stated separately.