Level-zero quaternionic subgroup conjecture for the Burau representation modulo $p$

Let $p$ be a prime satisfying $p\equiv1$ (mod 6). Let $\mathcal{Q}_{p}[0]$ be the quaternionic subgroup of level $0$, let $\mathcal{K}_{p}$ be the relevant free group, let $\mu_p$ be the map sending $\mathcal{K}_{p}$ into the ambient group, let $\widehat{\Lambda_p}[0]$ be the level-zero map, and let $\epsilon_p$ be the map whose kernel is the indicated free subgroup. Level-zero quaternionic subgroup conjecture. The image group $\mu_p(\mathcal{K}_p)$ is contained in $\mathcal{Q}_p[0]$, there are no additive generators in the rewritten words of the images of the generators of $\mathcal{K}_p$ under $\mu_p$, and

$$\widehat{\Lambda_p}[0]\bigl(\mu_p(\mathcal{K}_p)\bigr)\cong C_3.$$

In particular,

$$\mu_p\bigl(\ker(\epsilon_p)\bigr)\subseteq\ker\bigl(\widehat{\Lambda_p}[0]\bigr).$$

For primes $p>3$, the source notes that order-$3$ elements of $\mathcal{K}_p$ are conjugates of multiplicative generators, but the further structural assertion remains unproved.