Let $V$ be a Yetter-Drinfeld module over a Hopf algebra $H$ with invertible antipode. Recall that $V$ is a braided vector space with braiding $\Psi\colon V\otimes V\to V\otimes V, v\otimes w\mapsto v_{(-1)}w\otimes v_{(0)}$. The braid group $\mathbb{B}_r$ with $r-1$ generators $\sigma_1,\ldots,\sigma_{r-1}$ acts on $V^{\otimes r}$ by $\sigma_ix=\Psi_{i,i+1}x$ where $x\in V^{\otimes r}$ and where $\Psi_{i,i+1}$ is the braiding of the $i$th and $(i+1)$th tensor factor.

**Question**. Does there exist a Yetter-Drinfeld module $V$ over some Hopf algebra $H$ with invertible antipode such that the action of $\mathbb{B}_r$ on $V^{\otimes r}$ is faithful for some $r>1$?

A proof in special cases, e.g. $H=\mathbb{C}\Gamma$ for a group $\Gamma$, is welcome. I am interested in the ground field $\mathbb{C}$ and don't want positive characteristic examples.

**If you know related literature, this is accepted as an answer as well.**

**EDIT**. What I proved so far and which leads me to this question is as follows:

For each $r>1$, there exists a Yetter-Drinfeld module $V$ over $\mathbb{S}_{r+1}$ such that $t(\sigma)x=x$ for all $x\in V^{\otimes r}$ and some $\sigma\in\mathbb{S}_r$ implies $\sigma=1$. Here, $t\colon\mathbb{S}_r\to\mathbb{B}_r$ is the Matsumoto section.