  We prove that for $n\geq 2$, the size $b(t_n)$ of the smallest bidirectional
scheme for the $n$th Thue--Morse word $t_n$ is $n+2$. Since Kutsukake et al.
[SPIRE 2020] show that the size $\gamma(t_n)$ of the smallest string attractor
for $t_n$ is $4$ for $n \geq 4$, this shows for the first time that there is a
separation between the size of the smallest string attractor $\gamma$ and the
size of the smallest bidirectional scheme $b$, i.e., there exist string
families such that $\gamma = o(b)$.
