Asymptotic improvement factor for perfect matchings: show C_k^∞ > 1

Prove that for every integer k ≥ 4 (and, more generally, for fixed k ≥ 2), the asymptotic improvement factor C_k^∞ := liminf_{n→∞} min_G perm(G)/B_S(k,n) for k-regular bipartite graphs G on 2n vertices strictly exceeds 1, where B_S(k,n) = ((k-1)^{k-1}/k^{k-2})^n is Schrijver’s lower bound.

Background

This problem concerns the gap between the true minimum number of perfect matchings in k-regular bipartite graphs and Schrijver’s celebrated lower bound. The paper shows that Schrijver’s bound equals the Bethe approximation and proves strict improvements in certain cases (e.g., for k=3), but a general separation remains unknown.

The authors define the asymptotic improvement factor C_k^∞ as the liminf over graph size of the minimum ratio perm(G)/B_S(k,n). Establishing C_k^∞>1 would certify a uniform asymptotic improvement over Schrijver’s bound for fixed degree k.