Let $X$ and $Y$ be smooth second-countable Hausdorff manifolds equipped with fixed smooth positive densities, and let $S \subset X \times Y$ be an embedded submanifold. Let
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\begin{align*}
p_X:X \times Y \to X
\end{align*}
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and
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\begin{align*}
p_Y:X \times Y \to Y
\end{align*}
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denote the coordinate projections. Let $K \in I^m_{\mathrm{loc}}(X \times Y;S)$ be a distributional density conormal to $S$. Assume that $K$ is properly supported over $X$ in the following sense: for every compact set $C \subset X$, the set