If $X^n$ is an $E$-valued process and $f\in C_0(E)$ with $\text E\left[(fh)(X^n_t)\right]\xrightarrow{n\to\infty}0$ for all $h\in C_c(E)$, then $f=0$?

Question

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $E$ be a locally compact Hausdorff space, $(X^n_t)_{t\ge0}$ be an $E$-valued process on $(\Omega,\mathcal A,\operatorname P)$ for $n\in\mathbb N$ and $f\in C_0(E)$ with $$\operatorname E\left[f(X^n_t)\prod_{i=1}^kh_i\left(X^n_{t_i}\right)\right]\xrightarrow{n\to\infty}0\tag1$$ for all $k\in\mathbb N_0$, $0\le t_1<\cdots<t_k\le t$ and $h_1,\ldots,h_k\in C_c(E)$.

Are we able to conclude $f=0$?

My idea is that by $(1)$, $$\operatorname E\left[f(X^n_t)h(X^n_t)\right]=\int fh\:{\rm d}(X^n_t)_\ast\operatorname P\xrightarrow{n\to\infty}0\tag2$$ for all $h\in C_c(E)$. Is this sufficient to conclude? Which argument do we need?