It seems that Theorem 2 in the linked paper readily extends to variables taking values in polish spaces. The needed observation is that if the underlying random variables $X_1,\ldots,X_n$ take values in a Polish space $\Upsilon$, then the following equivalence holds:

$X_J$ and $X_K$ are conditionally independent given $X_L$ iff for all Borel functions $f:\Upsilon \to {\mathbb Z}$ with finite range, $f(X_J)$ and $f(X_K)$ are conditionally independent given $f(X_L)$.

Here $f(X_L)$ is shorthand for the sequence $\{f(x_\ell)\}_{\ell \in L}$.

It seems that Theorem 2 in the linked paper readily extends to variables taking values in polish spaces. The needed observation is that if the underlying random variables $X_1,\ldots,X_n$ take values in a Polish space $\Upsilon$, then the following equivalence holds:

$X_J$ and $X_K$ are conditionally independent given $X_L$ iff for all Borel functions $f:\Upsilon \to {\mathbb Z}$ with finite range, $f(X_J)$ and $f(X_K)$ are conditionally independent given the $\sigma$-field determined by $X_L$.

Here $f(X_L)$ is shorthand for the sequence $\{f(x_\ell)\}_{\ell \in L}$.