Kong's modulo-$8$ conjecture for near-perfect matchings of odd square grids

Let $a(2k+1)$ be the total number of near-perfect matchings of the $(2k+1)\times(2k+1)$ grid, and write

$$a(2k+1)=2^k c_k.$$

Kong's modulo-$8$ conjecture. For all $k$, the integer $c_k$ is congruent to $1$ modulo $8$.

This is presented as a further conjecture of Kong after the paper proves only that $c_k$ is odd; its status is not resolved in the supplied text.