Weak virial positivity conjecture for random regular bipartite graphs

Let $G$ be an $r$-regular bipartite graph with $v=2n$ vertices, let $m_i$ denote the number of $i$-matchings, and define

$$u(i)=-\ln(i!m_i).$$

For the finite-difference operator defined by $\Delta u(i)=u(i+1)-u(i)$, consider the probability over the relevant random regular bipartite graphs. Weak virial positivity conjecture. For each $i$ and $k\geq 2$,

$$\operatorname{Prob}(\Delta^k u(i)\geq 0)\xrightarrow[n\to\infty]{}1.$$

This is a weaker property than requiring every graph to satisfy Virial Positivity. The paper states that it proves a related result, so the precise scope of what remains conjectural should be checked against the subsequent theorem or proposition.