Let $X, X_1, X_2, \ldots $ be a sequence of $\mathbb{R}^d$-valued random variables defined on a common probability space $(\Omega, \mathscr{F}, \mathbb{P})$ such that the pairs
$$\tag{1}(X_n,X) \quad \text{ admit a Lebesgue density }\quad \chi_n\in C(\mathbb{R}^{2d})\quad\text{ for each } n\in\mathbb{N},$$ and such that for each $(\emptyset\neq)\,U\subseteq\mathbb{R}^d$ open there is $\Omega_U\in\mathscr{F}$ with $\mathbb{P}(\Omega_U)>0$ and $N= N_U\in\mathbb{N}$ such that $$\tag{2}X(\omega), (X_n(\omega))_{n\geq N}\in U\qquad\text{ for each }\ \omega\in\Omega_U.$$
We call a function $\chi\in C(U\times U)$ ($U\subseteq\mathbb{R}^d$) factorizable if $\chi\equiv\chi(x,y)=\chi_1(x)\cdot\chi_2(y)$ for some $\chi_1, \chi_2\in C(U)$. I was wondering about the following question.
Question: Is there a (reasonably weak) form of convergence $X_n\rightarrow X$ which guarantees that
$$\tag{3}\forall\, (\emptyset\neq)\, U \text{ open } \ : \ \left.\chi_n\right|_{U^{\times 2}} \ \text{ is not factorizable for infinitely many } \ n\in\mathbb{N}?$$