Vanishing conjecture for filtration quotients of connected graph diagrams

Let $\hat {\cal B}_{m}^{\,u}$ be the diagram space with filtration quotients ${\cal G}_{f,o,e}^{\,}\,\hat {\cal B}_{m+2f}^{\,u}$, where $f,o,e$ are the filtration parameters. Vanishing conjecture. For

$$m>u+1>0\qquad\text{and}\qquad m>5\left(\frac{u}{2}-1\right),$$

the filtration quotients ${\cal G}_{f,o,e}^{\,}\,\hat {\cal B}_{m+2f}^{\,u}$ are trivial for all possible parameters $f,o,e$. If true, the paper states that this would imply quadratic growth in $m$ for fixed $u$, together with the displayed dimension bounds that follow from Theorem 2. The conjecture is motivated by the experimental calculations but is not proved.