Using the Chen-Stein method, one can bound the total variation distance between a sum of possibly dependent Bernoulli random variables $W=\sum_{i=1}^n X_i$ and a Poisson distribution using only the first and second moments of the $X_i$. For example, see Two Moments Suffice for Poisson Approximations, by Arratia et al..

Is there an analogous bound for the total variation distance in the multivariate case? For example, if

\begin{equation*}
W_1 : = \sum_{i=1}^n X_i, \ \ \
W_2 : = \sum_{j=1}^n Y_j.
\end{equation*}
are sums of (not necessarily independent) Bernoulli random variables, then can we bound $d_{TV}(\mathcal{L}(W_1, W_2), \mathcal{L}( Poi(E[W_1]), Poi(E[W_2]))$, using the first and second moments of $X_i$ and $Y_j$ (these poisson random variables are independent)$? In the literature, I've only found bounds for the case where these Bernoulli random variables are presumed to be independent.

EDIT: Whoops. I should have read through the paper that I cited. See Theorem 2 in Arratia et al.