Squier's conjecture on Burau at roots of unity 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.
 A: Presumably, this conjecture appears from the observation that the suitable powers of the standard generator $\sigma_1$ lies in the kernel, although it looks that Squier missed other obvious elements of kernels as you mentioned.
I mention the paper by Funar and Kohno
On Burau’s representations at roots of unity.
https://link.springer.com/article/10.1007/s10711-013-9847-0
where they proved a strengthened form of another Squier's conjecture (C2) that appeared in the same Squier's paper. The paper also contains several discussions and results on Squire's conjectures.
It seems that a more suitable formulation of Squier's conjecture is as follows.
For $N=(n_1,\ldots,n_{k-1})$, let $B_k\{N\}$ be the subgroup generated by
$\sigma_1^{2n_1}, (\sigma_1\sigma_2)^{3n_2}, \ldots, (\sigma_1\sigma_2\cdots \sigma_{k-1})^{kn_{k-1}}$.
Geometrically, the group $B_k\{n\}$ is the group generated by $n_i$-th powers of the Dehn twists along simple closed curves in $n$-punctured disk $D_n$
enclosing $i+1$ puncture points.
When $N=\{n,n,n,\ldots\}$, we simply write $B_k\{N\}$ by $B_k\{n\}$.
Modified Squier's conjecture, weak version
If $q$ is a primitive root of unity then the kernel of Burau representation $\beta_{-q}$ is equal to $B_k\{N\}$ for suitable $N$.
Modified Squier's conjecture, strong version
If $q$ is a primitive root of unity then the kernel of Burau representation $\beta_{-q}$ is equal to $B_k\{n\}$ for some $n$.
If the Modified conjecture (strong version) is true for infinitely many distinct primitive root of unity, then the Burau representation is faithful.
This follows from Theorem 2.1 of Funar-Kohno's paper that asserts that the intersection of $B_k\{n\}$ over an infinite set of integers $n$ is trivial.
Thus for $k>4$, the Modified conjecture (strong version) can be true for only finitely many primitive root of unities.
On the other hand, Corollary 3.2 of Funar-Kohno's paper shows that if $q$ is a primitive $2n$-th root of unity for odd $n$ and $2n\geq7$, then modified conjecture (strong version) is true:
Corollary 3.2 says that the kernel $K$ of the Burau representation of 3-braids at $-q$ is normally generated by $\sigma_1^{2n}, \sigma_2^{2n} ,(\sigma_1^{2}\sigma_2^{2})^{n}$. Since
$ (\sigma_1\sigma_2)^{3n}=(\sigma_1^{2}(\sigma_2\sigma_1^{2}\sigma_2))^{n} = \sigma_1^{2n} (\sigma_2 \sigma_1^{2}\sigma_2)^{n}$
and $(\sigma_2 \sigma_1^{2}\sigma_2)^{n}$ is conjugate to $(\sigma_1^{2}\sigma_2^{2})^{n}$, the kernel $K$ coincides with $B_3\{n\}$.
These results say that the modified conjecture (weak or strong version) is meaningful and subtle for $k=4$ case. I feel it reasonable to investigate the modified conjectures as a possible approach of the faithfulness problem of Burau representation of 4-braids.
