AGV mechanism and Individual rationality
The simplest example I could cook up. I might have messed up somewhere.

Consider the provision of a public good with two agents, $n=2$. Let the type space be such that $\theta \in \{0,1\}$, and both types are equally likely. Either the good is provided, $k=1$, and total cost $c=2/3$ occurs, or the good is not provided, $k=0$, which is costless.
Let 
$$v_i (k; \theta_i) = k  (\theta_i - \frac{c}{2}).$$
That is, either the good is not provided and nobody pays or it is provided and both share the cost. Of course transfers can be used to compensate.
It is efficient to provide if at least one agent values the good, i.e., has a type of 1. AGV leads to this efficient rule $k^* (1,1)=k^* (0,1)=k^* (1,0)=1$ and $k^* (0,0)=0$.

Then,
$$\xi_i (1) = \frac{1}{2} \left( v_j(k^*(1,0);0) + v_j(k^*(1,1);1) \right)
= \frac{1}{2} (0+1 - c) = \frac{1}{6},$$
$$\xi_i (0) = \frac{1}{2} \left( v_j(k^*(0,0);0) + v_j(k^*(0,1);1) \right)
= \frac{1}{2} (0+1- c/2) = \frac{1}{3},$$

Now, suppose $(\theta_i,\theta_j)=(0,1)$. Then,
$$v_i(1;0)+\xi_i(0) = 0 - \frac{1}{3}  + \frac{1}{3}= 0 &lt; \xi_j (1) = \frac{1}{6}.$$
So, we don't have ex-post IC in that case.