Let $\{X_t\}_{t\in N}\subseteq R^{n\times n}$ be a Markov chain defined by matrix product $X_{t} = G_1 \dots G_t$ where $G_i$s are iid Gaussian matrices $G_1,\dots,G_t\sim N(0,1)^{n\times n}$. Is this chain ergodic? In other words, can we find a coupling for two copies of this chain $\{X_t\}_{t\in N}$ and $\{Y_t\}_{t\in N}$ such that $Pr(X_t\neq Y_t) \le \alpha^t TV(X_0,Y_0)$ for some $\alpha<1$ and some "nice" initial distributions of the two chains (eg. the scale of initial matrices is similar $E \|X_0\|_F=E \|Y_0\|_F$)?

Intuitively this coupling should be plausible: Since Gaussian matrices $G_i$s are rotation invariant, we can couple the $X_t$ and $Y_t$ by optimising for rotation. But I come short of making these intuitions any more rigorous. I've looked up literature on random matrices but can't find anything relevant.