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 $x$, and for arbitrary Borel sets $A_i \subset C[0, T]$
$$\nu_x(\bigcap_i \, \{X^i \in A_i\}) = \ \prod_i \nu_x (X^i \in A_i).$$
and so
$$\mathbb P (\nu_x(\bigcap_i \, \{X^i \in A_i\}) = \ \prod_i \nu_x (X^i \in A_i))$$ $$= \int_{\mathbb R^n} 1_{\{\nu_x(\bigcap_i \, \{X^i \in A_i\}) = \ \prod_i \nu_x (X^i \in A_i)\}} \, d\mu_X (x)$$ $$= \int_{\mathbb R^n} 1 \, d\mu_X (x)$$ $$ = 1.$$
And so we conclude conditional independence of the processes $X^i$ as desired.