Given an integer $n\ge 1$, let $ \textstyle \Delta:=\left\{x \in [0,1]^n: \sum_{i\le n}x_i=1\right\} $ be the $n-1$-dimensional simplex. Consider a random variable $ X: \Omega\to \Delta $ (on a measurable space $\Omega$) with the uniform distribution on $\Delta$, let us say $\mathrm{Pr}$ (in other words, this is the Dirichlet distribution).
At this point, fix reals $a_1,b_1,a_2,b_2,\ldots,a_n,b_n \in [0,1]$ such that $\sum_{i\le n}a_i \le 1$ and $a_i< b_i$ for all $i$, and define $$ R:=\left\{x \in \Delta: a_i\le x_i\le b_i \text{ for all }i=1,\ldots,n\right\}. $$
Question. Is there a "reasonable" way to compute the expected value of $X$, given that $X \in R$? In other words, is it possible to compute (as function of the parameters $a_i,b_i$) the value of $$ \mathbb{E}[\,X\,|\,R\,]:=\frac{1}{\mathrm{Pr}(R)}\int_{\omega \in \Omega:\, X(\omega) \in R} X(\omega)\,\mathrm{Pr}(\mathrm{d}\omega)\quad? $$