Does there exist a probability distribution over the set $\{(x,y,z)\in[0,1]^3\colon x+y+z=3/2\}$ whose projection on each of the three coordinate axes is the uniform distribution over the interval $[0,1]$?
This is a restatement of what remains to show in order to make the previous answer quite complete. This restatement is posted separately to attract more attention to the problem.
Strassen's Theorem 7 on p. 433 for two Polish spaces $S$ and $T$ can be extended to "any finite number $S,T,\ldots,R$ of Polish spaces", as noted in the middle of p. 434 of Strassen's paper. In our case, for three Polish spaces we take three copies of the interval $[0,1]$. So, the above question can be restated as follows:
Is it true that for all continuous real-valued functions $f,g,h$ on $[0,1]$ we have $$\int_0^1(f+g+h)\le\sup E[f(X)+g(Y)+h(3/2-X-Y)],$$ where the supremum is taken over all random variables $X$ and $Y$ such that $0\le X\le1$, $0\le Y\le1$, and $1/2\le X+Y\le3/2$?
This question has now been answered, quite elementarily, by Ilya Bogdanov.
