Modified cluster-expansion limit conjecture for proportional permanents

Let $A$ be sampled from the regular zero-one matrix ensemble, let $E_1$, $E_2$, and $E$ be the expectations defined in the paper, and let $\hat T_i(n)$ be the modified cluster-expansion terms for $m=\alpha n$, where $\alpha$ is fixed.

Modified cluster-expansion limit conjecture. Using $E_1$, $E_2$, or $E$, the following limit exists:

$$\lim_{n\to\infty}\frac{1}{n}\hat T_i(n).$$

This is proposed as the proportional-$m$ analogue of the earlier cluster-expansion limit conjecture. The source offers it as a computer-supported conjecture and does not state a resolution.