Furter's Rigidity Conjecture for polynomial inverses

Let $m\in\mathbb N_+$ and let $a(X)\in{\mathbb C}[X]$ be a polynomial of degree at most $m+1$ such that $a(X)\equiv X\pmod{X^2}$. Let $a^{-1}(X)$ denote its formal inverse for composition. Furter's Rigidity Conjecture. If $m$ consecutive coefficients of $a^{-1}(X)$ vanish, then $a(X)=X$. Furter proved the cases $m=1$ and $m=2$, while the paper describes the case $m=3$ as more complicated; the general conjecture remains open.