(Weak) Convergence of stationary distributions under tightness

Question

For real-valued stochastic processes $X_n(\cdot),Y_n(\cdot)$ on $[0,\infty)$, suppose we know the following holds:

1. If the initial condition $X_n(0)-Y_n(0)\rightarrow 0$ in probability as $n\rightarrow \infty$, then $X_n(t)-Y_n(t)\rightarrow 0$ in probability as $n\rightarrow \infty$ (or, $X_n(\cdot)-Y_n(\cdot)$ weakly converges to zero uniformly on compact sets).
2. Further suppose there exist stationary distributions (in the sense of Markov process, or limiting distributions) $X_n(\infty),Y_n(\infty)$, and suppose we know that $\{X_n(\infty)\}_{n\geq1}$ is tight (i.e., stochastically bounded).

Question: Can we conclude that $X_n(\infty)-Y_n(\infty)\rightarrow 0$ in probability as $n\rightarrow \infty$?