Filtration-independence conjecture for connected graph diagram spaces

Let $\hat {\cal B}_{m}^{\,u}$ be the diagram space equipped with the filtration whose quotient at $(f,o,e)$ is ${\cal G}_{f,o,e}^{\,}\,\hat {\cal B}_{m}^{\,u}$. Let $T(m,u)$ be the set of triples defined by the nonnegative integer constraints in Proposition 1, and suppose $(f_i,o,e)\in T(m+2f_i,u)$ for $i=1,2$. Filtration-independence conjecture. There are isomorphisms

$${\cal G}_{f_1,o,e}^{\,}\,\hat {\cal B}_{m+2f_1}^{\,u}\cong {\cal G}_{f_2,o,e}^{\,}\,\hat {\cal B}_{m+2f_2}^{\,u}$$

for all integers $m,u,f_1,f_2,o,e$ satisfying these conditions. The conjecture expresses the observed independence of the filtration quotients from the number of free squares, after shifting the degree. The paper reports evidence from many situations and later computations, but gives no proof.