Congruence of the normalized square-grid matching count

Prove that for every nonnegative integer k, the integer c_k defined by a(2k+1)=2^k c_k is congruent to 1 modulo 8.

Background

The paper defines a(2k+1) as the number of near-perfect matchings of the (2k+1)×(2k+1) square grid. It proves that a(2k+1) is an odd multiple of 2^k, so a(2k+1)=2^k c_k for an odd integer c_k.

The authors identify Kong’s stronger conjecture that the odd factor c_k is always congruent to 1 modulo 8. They prove that this conjecture would follow if the number of near-perfect matchings with a hole on the central row or column, excluding the center, were always divisible by 2^{k+1}; the required divisibility had only been checked computationally for k≤16 in the provided text.