Conjecture on universal rational-function limits of cluster-expansion terms

Let $A$ be sampled from the ensemble of $n\times n$ zero-one matrices with every row and column sum equal to $r$, and let $E$ denote expectation in this ensemble. Let $E_1$ and $E_2$ be the two approximating expectations defined in the paper, and let $T_i(n)$ be the corresponding cluster-expansion terms. Define

$$h(i)=\begin{cases}i/2,&\text{if }i\text{ is even},\\(i+1)/2,&\text{if }i\text{ is odd}. \end{cases}$$

Universal-limit conjecture. The limits obtained using $E_1$, $E_2$, or $E$ are equal, and

$$\lim_{n\to\infty}\frac{1}{n}T_i(n)\equiv Q_i=\sum_k\frac{a_k(i)}{r^k},$$

with $i-1\geq k\geq h(i)$. The claim refines the preceding existence conjecture by prescribing the dependence on $r$. The paper gives computed examples for $Q_2$ through $Q_7$ but no proof or resolution.