The formal cell-zeta dimension conjecture

Let $n=\ell+3$, let $d_k$ be defined by

$$d_k=d_{k-2}+d_{k-3},\qquad d_0=1,\\ d_1=0,\qquad d_2=1,$$

and consider formal cell-zeta values on ${\mathfrak{M}}_{0,n}$ modulo all linear relations from dihedral and modular shuffle relations. Let $\mathfrak{F}=\mathbb{Q}[e_2]\oplus\mathfrak{L}$, where $\mathfrak{L}$ is the free Lie algebra generated by one element $e_{2n+1}$ in each odd degree. The formal cell-zeta dimension conjecture. The dimension of this quotient $\mathbb{Q}$-vector space is $d_\ell$; equivalently, the dual Lie algebra to the resulting co-Lie algebra after quotienting by products is isomorphic to $\mathfrak{F}$. The paper verifies this for $n\leq 9$ by direct calculation, but leaves the general assertion conjectural.