Let $\nu_x$ be the regular conditional probability associated with $X$, and $\mu_X$ the law of $X$ on $\mathbb R^n$ Under $\nu_x$, the $X^i_0$ are deterministic, and hence also the process $\eta_s$.

As such, each $X^i$ is a standard diffusion SDE driven by $B^i$ with non-random coefficients, for which it is known there is a strong solution. Thus each $X^i$ is $\sigma(B^i)$ measurable under $\nu_X$, almost surely.

Then for $\mu_X$ almost all $x$,

$$\mu_x \left (\bigcap_i \, \{X^i \in A_i\} \
\prod_i \nu_x (X^i \in A_i) \, \forall A_i \in \mathcal B(C[0, T])\right ) := \mu_x (E) = 1,$$

and so 

$$\mathbb P (\nu_x(E))$$
$$= \int_{\mathbb R^n} 1_E (x) \, d\mu_X (x)$$
$$= \int_{\mathbb R^n} 1 \, d\mu_X (x)$$
$$ = 1.$$

And so we conclude conditional independence of the processes $X^i$ as desired.