Full rank of the girth connecting homomorphisms

Prove that, for every loop order \(n\) and every homological degree \(k\), the connecting homomorphism \(d_{1,k}\) in the long exact sequence associated with the girth filtration of the zero-hair 3-edge-connected forested graph complex has full rank.

Background

The paper filters the 3-edge-connected forested graph complex by girth and obtains connecting homomorphisms $d_{1,k}$ between the homologies of the girth-one quotient and the subcomplex of girth at least two. Full rank of these maps would make the dimension estimate from the associated long exact sequence exact and could substantially reduce the computational cost of determining $H_k(Out(F_n))$. Numerical computations establish full rank for $n\leq 7$, motivating the stated conjecture for all loop orders.