Upper asymptotic matching conjecture

Let $\{G_k=(V_k,E_k)\}_{k\in\mathbb{N}}$ be a sequence of $r$-regular bipartite graphs with $\#V_k\to\infty$. Let $h_{\{G_k\}}(p)$ denote the associated monomer–dimer entropy, and let $h_{K(r)}(p)$ be the entropy for a countable disjoint union of copies of $K_{r,r}$. Upper asymptotic matching conjecture. For every $p\in[0,1]$,

$$h_{\{G_k\}}(p)\leq h_{K(r)}(p).$$

This is the asymptotic consequence of the upper matching conjecture and asserts that disjoint unions of complete bipartite graphs maximize the matching entropy among regular bipartite graph sequences. The supplied text gives no resolution status.