First: just a bit of background on copulas.
Suppose we have a pair of continuous random variables $Y_1, Y_2$ with distribution functions $F_1(y_1)=P(Y_1\leq y_1)$ and $F_2(y_2)=P(Y_2\leq y_2)$.
Let their joint distribution function $F(y_1, y_2) = P(Y_1\leq y_1, Y_2\leq y_2)$ be described by the copula $C$ with $F(y_1, y_2) = C(F_1(y_1), F_2(y_2))$.
The joint density is then given by $$ \begin{align} f(y_1, y_2) &=\frac{\partial^2}{\partial y_1 \partial y_2}F(y_1, y_2) \\ &=\underbrace{\left[\frac{\partial^2}{\partial u_1 \partial u_2}C(u_1, u_2)\right]_{u_1=F_1(y_1),\\ u_2=F_2(y_2)}}_{\equiv c(F_1(y_1), F_2(y_2))}\cdot f_1(y_1) f_2(y_2)\,, \end{align} $$ where $f_1,f_2$ are the marginal densities and $c$ is the copula density.
Now to the meat of the question. Let's introduce a third random variable $X$. Suppose we observe some $iid$ realizations of the vector $(X, Y_1)$ and of the vector $(X, Y_2)$ but crucially never the vector of all three, $(X, Y_1, Y_2)$.
Using these data, say we estimate some conditional densities, say $\hat f_1(y_1|x)$ and $\hat f_2(y_2|x)$. However, we have no direct information about the joint copula.
Suppose we want to estimate the joint conditional density $f(y_1, y_2|x)$. I would conjecture that the "best" we can do is something of the form $$ \hat f(y_1, y_2|x) \propto c(F_1(y_1), F_2(y_2)) \hat f_1(y_1|x) \hat f_2(y_2|x)\,. $$
This is of course an unnormalized density and does not in general have $\hat f_1(y_1|x)$ and $\hat f_2(y_2|x)$ as marginals. But it does feel like a "sensible" way of combining two conditional estimates given an unconditional copula.
Is there a more rigorous notion of "good" that this maximizes? Are there any estimators of this form in the existing literature?
Thanks in advance!