The graph positivity conjecture

Consider $r$-regular bipartite graphs with $v=2n$ vertices. Let $m_i$ be the number of $i$-matchings, and let $\overline{m}_i$ be the number of $i$-matchings in the complete graph on the same vertices. Define

$$d(i)=\ln\left(\frac{m_i}{r^i}\right)-\ln\left(\frac{\overline{m}_i}{(v-1)^i}\right),$$

and let $\Delta d(i)=d(i+1)-d(i)$. A graph satisfies graph positivity when $\Delta^k d(i)\geq 0$ for all meaningful $i$ and $k$. The graph positivity conjecture. For fixed $r$, as $n$ tends to infinity, the fraction of graphs that satisfy graph positivity approaches one. This conjecture is supported by computer evidence, including the reported behavior for small regular bipartite graphs; the supplied text gives no proof or disproof.