In Squier's short, yet influential, paper about the Burau representation, he made two conjectures that might have provided a proof for the faithfulness of the Burau representation (which we now know to be false by work of Bigelow, for instance).
Intro. The (reduced) Burau representation $\beta$ sends the Artin generators $\sigma_i$ for the braid group $B_n$ to the $(n-1)\times(n-1)$ matrix over the Laurent polynomials $\mathbb Z[t^\pm]$ given by$$\beta(\sigma_i) = \operatorname{Id}_{i-2} \oplus \begin{pmatrix} 1 & 0 & 0 \\ t & -t & 1 \\ 0 & 0 & 1 \end{pmatrix} \oplus \operatorname{Id}_{n-i-2}$$with a slight, easy modification in the $i=1$ and $i=n-1$ cases when this formula doesn't make sense. The powers of $\sigma_i$ have a nice formula that ends up being a bit cleaner to write if we substitute $t=-q$. We get$$\beta(\sigma_i^d) = \operatorname{Id}_{i-2} \oplus \begin{pmatrix} 1 & 0 & 0 \\ -q(1+q+\dotsb+q^{d-1}) & q^d & (1+q+\dotsb+q^{d-1}) \\ 0 & 0 & 1 \end{pmatrix} \oplus \operatorname{Id}_{n-i-2}$$
Squier's conjecture. In case $q$"specialized" to be a primitive $d$-th root of unity, you can see that $\beta(\sigma_i)$ has order $d$. Squier conjectured the following:
(C1) The kernel of the composite map $\beta_{-q}:B_n \xrightarrow{\beta} \operatorname{GL}_{n-1}(\mathbb Z[t^\pm]) \xrightarrow{t \mapsto -q} \operatorname{GL}_{n-1}(\mathbb C)$, when $q$ is a primitive $d$-th root of unity, is exactly equal to the normal subgroup $\langle\langle \sigma_i^d \rangle\rangle$.
My issue. Consider the full twist braid $\Delta^2=(\sigma_1\dotsb\sigma_{n-1})^n$ in $B_n$. You can show that $\beta(\Delta^2)=t^n\cdot \operatorname{Id}_{n-1}$. What power of $\Delta^2$ lies in $\ker \beta_{-q}$? Consider the case $n=3$, $d=6$. Then $\beta(\Delta^2) = t^3 \mapsto -q^3 = 1$ under the specialization at $-q$, so $\Delta^2 \in \ker \beta_{-q}$. But $\Delta^2 \notin \langle\langle \sigma_i^6 \rangle\rangle$! (For instance, forgetting one strand sends eleemnts of $\langle\langle \sigma_i^6 \rangle\rangle$ to 6-th powers in $B_2$, but $\Delta^2$ maps to $\sigma_i^2$.) It shouldn't be hard to find lots of examples like this, powers of $\Delta^2$ that easily lie in $\ker \beta_{-q}$ but don't lie in $\langle\langle \sigma_i^d \rangle\rangle$.
Question. Did Squier just overlook this seemingly simple counterexample to his conjecture? Does the conjecture seem more reasonable if we replace $\langle\langle \sigma_i^d \rangle\rangle$ with $\langle \Delta^{2k} \rangle\langle\langle \sigma_i^d \rangle\rangle$ for some power $k$ depending on $d$?
Squier, Craig C., The Burau representation is unitary, Proc. Am. Math. Soc. 90, 199-202 (1984). ZBL0542.20022.