Let $P$ be a probability distribution on the product of Polish spaces $X_1\times \cdots \times X_N$. In particular, $X_N=X_j$. Suppose the marginal distribution of $P$ on $(X_i)$ are $(\mu_i)$, where $(\mu_i)$ are absolutely continuous with respect to Lebesgue measure. In particular, $\mu_N=\mu_j$.
My question is if the projection of $P$ onto 2-dimension space is the same regardless of $j$ or $N$. That is, if for any $i\neq j$ and $i\neq N$, we have: $$P(X_1\times\cdots\times A_i\times\cdots\times A_j\times \cdots\times X_N)=P(X_1\times \cdots\times A_i\times \cdots\times X_{N-1}\times A_j)$$ for all Borel sets $A_i\subset X_i$ and $A_j\subset X_j$.
This question might be related with if one can uniquely determine the 2-joint distribution from two marginals.
Let $j=2$ an $N=3$. Suppose all the $X_i$ are $[0,1]$ with Lebesgue measure and let $P$ by the distribution of $U,U,V$ where $U,V$ are independent uniform variables in $[0,1]$. In other words, $P$ is the image of Lebesgue measure on the square $[0,1]^2$ under the map $(x,y) \mapsto (x,x,y)$. You get a negative answer to your question by taking $A_1$ and $A_2$ to be disjoint intervals.