Aldous-type spectral gap results for the complete monomial group

Let us consider the continuous-time random walk on $G\wr S_n$, the complete monomial group of degree $n$ over a finite group $G$, as follows: An element in $G\wr S_n$ can be multiplied (left or right) by an element of the form \begin{itemize} \item $(u,v)_G:=(\mathbf{e},\dots,\mathbf{e};(u,v))$ with rate $x_{u,v}(\geq 0)$, or \item $(g)^{(w)}:=(\dots,\mathbf{e},\hspace*{-0.65cm}\underset{\substack{\uparrow\\w\text{th position}}}{g}\hspace*{-0.65cm},\mathbf{e},\dots;\mathbf{id})$ with rate $y_w\alpha_g\; (y_w> 0,\;\alpha_g=\alpha_{g^{-1}}\geq 0)$, \end{itemize} such that $\{(u,v)_G,\;(g)^{(w)}:x_{u,v}>0,\;y_w\alpha_g>0,\;1\leq u<v\leq n,\;g\in G,\;1\leq w\leq n\}$ generates $G\wr S_n$. We also consider the continuous-time random walk on $G\times\{1,\dots,n\}$ generated by one natural action of the elements $(u,v)_G,1\leq u<v\leq n$ and $(g)^{(w)},\;g\in G,1\leq w\leq n$ on $G\times\{1,\dots,n\}$ with the aforementioned rates. We show that the spectral gaps of the two random walks are the same. This is an analogue of the Aldous' spectral gap conjecture for the complete monomial group of degree $n$ over a finite group $G$.