Trotter-Kato approximation theorem for transition semigroups of Feller processes

Question

Let

• $E$ be a locally compact separable metric space;
• $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroup on $C_0(E)$ with generator $C_0(E)$;
• $D$ be a core of $(\mathcal D(A),A)$;
• $E_n$ be a metric space for $n\in\mathbb N$;
• $(T_n(t))_{t\ge0}$ be a strongly continuous contraction semigroup on the space $\mathcal F_b(E_n)$ of bounded Borel measurable functions from $E_n$ to $\mathbb R$ equipped with the supremum norm with generator $(\mathcal D(A_n),A_n)$ for $n\in\mathbb N$;
• $\pi_n:E_n\to E$ be continuous and $$\iota_nf:=f\circ\pi_n\;\;\;\text{for }f\in C_0(E)$$ for $n\in\mathbb N$;
• $(\Omega,\mathcal A,\operatorname P)$ be a probability space;
• $(X_t)_{t\ge0}$ be an $E$-valued càdlàg process on $(\Omega,\mathcal A,\operatorname P)$ with $$\operatorname E\left[f(X_t)\mid(X_r)_{r\le s}\right]=(T(t-s)f)(X_s)\;\;\;\text{almost surely}\tag1$$ for all $f\in C_0(E)$ and $t\ge s\ge0$; and
• $(Y_t^n)_{t\ge0}$ be an $E_n$-valued càdlàg process on $(\Omega,\mathcal A,\operatorname P)$ with $$\operatorname E\left[f(Y^n_t)\mid(Y^n_r)_{r\le s}\right]=(T_n(t-s)f)(Y^n_s)\;\;\;\text{almost surely}\tag2$$ for all $f\in C_b(E_n)$ (the subspace of $\mathcal F_b(E_n)$ of continuous functions) and $$X^n:=\pi_n\circ Y^n$$ for $n\in\mathbb N$.

Consider the following claims:

1. If $f\in C_0(E)$ and $t>s\ge 0$, there is a $G_n\in\mathcal B(E_n)$ for $n\in\mathbb N$ with $$\lim_{n\to\infty}\operatorname P\left[Y^n_s\in G_n\right]=1\tag3$$ and $$\lim_{n\to\infty}\sup_{x\in G_n}\left|(T_n(t-s)\pi_nf)(x)-(\pi_nT(t)f)(y)\right|=0.\tag4$$
2. If $f\in D$ and $t\ge0$, there is a $G_n\in\mathcal B(E_n)$ and a $f_n\in\mathcal D(A_n)$ for $n\in\mathbb N$ with $$\lim_{n\to\infty}\operatorname P\left[\forall s\in[0,t]:Y^n_s\in G_n\right]=1,\tag5$$ $\sup_{n\in\mathbb N}\left\|f_n\right\|_\infty<\infty$ and $$\lim_{n\to\infty}\left(\sup_{x\in G_n}|f_n(x)-(\iota_nf)(x)|+\sup_{x\in G_n}|(A_nf_n)(x)-(\iota_nAf)(x)|\right)=0.\tag6$$

Are we able to show that 2. implies 1.?

In the usual setting of the Trotter-Kato approximation theorem, we're able to conclude something stronger than 1. from 2.. The problematic thing here are the sets $G_n$ and that we only know $(T_n(t))_{t\ge0}$ is contractive (i.e. operator norms at most 1) on the whole space.