Extension conjecture for the exceptional Markov trace

Let $t_3^0$ be the Markov trace on the specialized algebra described by the conditions $t_3^0(1)=t_3^0(s_1)=0$ and $t_3^0(s_1s_2)=1$. Markov-trace extension conjecture. The trace $t_3^0$ can be extended to a Markov trace on the tower $(F_n)$. The extension exists through $F_4$ by computer calculation, but whether it extends to a genuine Markov trace on the full tower remains open; any obstruction must occur on at least five strands.