A recursive upper-bound conjecture for graph partition weights

Let $\Pi(n,r)$ denote the quantity counting the relevant partitions or graph contributions, and let $C_{n,r}$ be the geometric factor associated with graphs having $r$ vertices and $n$ edges. Define

$$\widetilde{\Pi}(n,r)=C_{n,r}\,\Pi(n,r).$$

Recursive upper-bound conjecture. There exists a geometric constant $C<\infty$ such that

$$\widetilde{\Pi}(n,r)\leq \widetilde{\Pi}(n-1,r-1)+C\,\widetilde{\Pi}(n-1,r)$$

for $n,r\in\mathbb{N}$ with $n\geq r$, and with

$$\widetilde{\Pi}(r,r)=C^r.$$

The proposed bound is intended to exploit the decay of the edge factors $\Phi_{b,b'}$ in the graph expansion and thereby improve the factorial growth of the cruder estimate. The source presents this as a proposed recursive bound; no resolution is supplied here.