Universal dimension bound for primitive graph $c_2$-invariants

Let $G$ be a primitive graph with first Betti number $h_1(G)$, and let $\dim(c_2(G))$ denote the dimension of a minimal model of its $c_2$-invariant. Universal dimension-bound conjecture.

$$\dim(c_2(G))\leq2h_1(G)-11.$$

The bound is stated to be sharp at loop order $8$ for $P_{8,41}$, but its general validity remains open.