Conjecture on Brownian motion and cutoff on the reflection quantum group H_N^{\infty+}

Let $H_N^{\infty+}$ be the infinite reflection quantum group, let $h$ denote its Haar state, and let $\varphi$ denote the standard Brownian motion on $H_N^{\infty+}$ viewed as a Lévy process. Write $d_{\mathrm{TV}}$ for total variation distance and set

$$d_N(t):=d_{\mathrm{TV}}(\varphi_t,h).$$

Brownian motion and cutoff conjecture. For every $t\geq 0$, the state $\varphi_t$ is not absolutely continuous with respect to the Haar state. Moreover,

$$\liminf_{N\to\infty}d_N\left(N\ln N+cN\right)\geq\frac{e^{-c}}{1+e^{-c}}=f_\infty(c),\qquad c\in\mathbf{R}.$$

These assertions concern the singularity and the large-$N$ total-variation behavior of Brownian motion on $H_N^{\infty+}$; the parser supplies no evidence resolving them, so their status remains open.