Simplifying assumption for regular vine distributions

Question

I am trying to understand vine copulas and specifically, conditional bivariate copulas. Let $c_{X_1,X_3|X_2}(\cdot,\cdot | \cdot)=:c$ be the probability density function of a bivariate conditional copula. I think I understand what $$ c_{X_1,X_3|X_2}(F_{X_1|X_2}(x_1|x_2),F_{X_3|X_2}(x_3|x_2)) $$ means. For any given value $x_2$ of the random variable $X_2$, we can look at the joint distribution of the random vector $(X_1,X_3)$. It is $F_{X_1,X_3|X_2}(x_1,x_3|x_2)$. We can express this joint distribution by

• the marginal distributions $F_{X_1}(x_1)$, $F_{X_2}(x_2)$ and $F_{X_3}(x_3)$,
• the pairwise copulas $c_{X_1,X_2}(F_{X_1}(x_1),F_{X_2}(x_2))$ and $c_{X_3,X_2}(F_{X_3}(x_3),F_{X_2}(x_2))$ and
• the pairwise conditional copula $c$ specified above: \begin{aligned} f_{X_1,X_3|X_2}(x_1,x_3|x_2) = &f_{X_1}(x_1) \cdot f_{X_2}(x_2) \cdot f_{X_3}(x_3) \\ \cdot \ &c_{X_1,X_2}(F_{X_1}(x_1),F_{X_2}(x_2)) \cdot c_{X_3,X_2}(F_{X_3}(x_3),F_{X_2}(x_2)) \\ \cdot \ &c_{X_1,X_3|X_2}(F_{X_1|X_2}(x_1|x_2),F_{X_3|X_2}(x_3|x_2)) \end{aligned}

(see e.g. Dissmann et al. (2013) Theorem 2.5).

There is an assumption$\color{red}{^*}$ that the pairwise conditional copula function does not vary with the values of $X_2$ (according to Czado, 2019) and is independent of the conditioning variables in higher-dimensional case (according to Dissmann et al., 2013). As I understand, this is a genuine assumption and it need not hold in general. E.g. I can imagine a situation where the following two hold simultaneously:

• $(X_1,X_3)|X_2=\color{red}{1} \sim N\left( \mu, \Sigma \right)$ and
• $(X_1,X_3)|X_2=\color{red}{2} \sim t\left( \mu, \Sigma, \text{df}=5 \right)$ where $t$ is a bivariate Student-$t$ distribution.

Questions

1. If this is indeed possible, are there any examples (from the literature or made up on the spot) where this is problematic? E.g. may this realistically lead to severe underestimation of tail risk (e.g. the expected shortfall) of returns on an asset portfolio$\color{red}{^{**}}$?
2. In a $k$-dimensional case (generalizing the current example of $k=3$), does the assumption allow the conditional distributions of $(X_i,X_j)|X_l$ and $(X_i,X_j)|X_m$ to differ as long as $l\neq m$? If not, then is it the same as the unconditional distribution of $(X_i,X_j)$?
3. Can we factor any multivariate distribution into an R-vine when restricted by the assumption?
4. Can we factor any multivariate distribution into an R-vine absent the assumption?
   In other words, can any multivariate copula be factored into an R-vine?
   (Think Sklar's theorem adapted for R-vines instead of a multivariate copula.)

$\color{red}{^*} \color{white}{^*}$ See e.g. Remark 5.12 in Czado (2019) p. 103, Dissmann et al. (2013) p. last paragraph of p. 55 and the references therein.
$\color{red}{^{**}}$ Returns on a portfolio is a random variable that is a weighted sum of a set of random variables, namely, returns on individual assets. The joint distribution of returns on the individual assets is modelled by an R-vine, and the distribution of a weighted combination (the portfolio returns) is inferred from there.

References

• Czado, C. (2019). Analyzing dependent data with vine copulas. Lecture Notes in Statistics, Springer.
• Dissmann, J., Brechmann, E. C., Czado, C., & Kurowicka, D. (2013). Selecting and estimating regular vine copulae and application to financial returns. Computational Statistics & Data Analysis, 59, 52-69.