The Awesome conjecture

Let $r$ and $j$ be parameters, let $z_i$ be positive integers, and let $c_i$ be coefficients. Define

$$F=\sum_{s\geq 0}\frac{a_s(r,j)}{n^s}+\sum_i c_i j(j-1)\cdots(j-z_i+1)\frac{1}{n^{z_i}r^{z_i}}\sum_{s\geq 0}\frac{a_s(r,j-z_i)}{n^s},$$

with $a_0=1$. The Awesome conjecture. The coefficients of $\ln(F)$ satisfy

$$[j^k n^{-h}]\ln(F)=0,\qquad k\geq h+2,$$

and

$$[j^{h+1}n^{-h}]\ln(F)=\frac{1}{(h+1)h}\left(\frac{1}{r^h}-2\right).$$

Here $[j^k n^{-h}]$ denotes coefficient extraction in the formal expansion in $j$ and $n^{-1}$. The conjecture is presented as needed for a proof of weak graph positivity; the supplied text says that it remains to be proved.