Conjecture on the asymptotic $k$-path partition function

For $c$, $\beta$, and \frac{1}{N^2}\frac{1}{2}} as in the preceding definitions, let $R^-_c(]\alpha,\beta)$ and $R^+_c(]\alpha,\beta)$ denote the lower and upper limits of the scaled expected logarithm of the $k$-path partition function, and let $f_c(\beta)$ be the corresponding free-energy derivative at zero path density. Asymptotic-partition-function conjecture. The lower and upper limits $R^-_c(]\alpha,\beta)$ and $R^+_c(]\alpha,\beta)$ agree, and moreover

$$\left.\frac{d}{d\alpha}R^+(c,\alpha;\beta)\right|_{\alpha=0}=f_c(\beta).$$

This is proposed as a stronger result than the preceding bounds, but the supplied text gives no resolution status.